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Applications of molecular dynamics techniques and spectroscopic theories to aqueous interfaces

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Applications of molecular dynamics techniques and spectroscopic theories to aqueous interfaces
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Green, Anthony
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Water
Azobenzene
Carbon tetrachloride
Silica
Aqueous interface
Molecular dynamics
SFG
SFVS
Nonlinear spectroscopy
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Abstract:
ABSTRACT: The primary goal of spectroscopy is to obtain molecularly detailed information about the system under study. Sum frequency generation (SFG) vibrational spectroscopy is a nonlinear optical technique that is highly interface specific, and is therefore a powerful tool for understanding interfacial structure and dynamics. SFG is a second order, electronically nonresonant, polarization experiment and is consequently dipole forbidden in isotropic media such as a bulk liquid. Interfaces, however, serve to break the symmetry and produce a signal. Theoretical approximations to vibrational spectra of O-H stretching at aqueous interfaces are constructed using time correlation function (TCF) and instantaneous normal mode (INM) methods. Detailed comparisons of theoretical models and spectra are made with those obtained experimentally in an effort to establish that our molecular dynamics (MD) methods can reliably depict the system of interest. The computational results presented demonstrate the potential of these methods to accurately describe fundamentally important systems on a molecular level.
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Dissertation (PHD)--University of South Florida, 2010.
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by Anthony Green.
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Applications of Molecula r Dynamics Techniques and Spectroscopic Theories to Aqueous Interfaces by Anthony J. Green A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Chemistry College of Arts and Sciences University of South Florida Major Professor: Brian Space, Ph.D. Randy Larsen, Ph.D. Milton Johnston, Ph.D. Preston B. Moore, Ph.D. Date of Approval: July 14, 2010 Keywords: water, azobenzene, carbon tetrachloride, silica, aqueous interface, molecular dynamics, SFG, SFV S, nonlinear spectroscopy Copyright 2010, Anthony J. Green

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Dedication This work is dedicated to Ella Green, L ynn Nicholls, Clyde Brown, Marion Fiorillo, Michael Frost, and Jamie Starkweather. I miss you all.

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Acknowledgements First, I would like to express my eterna l gratitude to my amazing wife, Lindsay. Her love and patience seem to have no end. The unwavering support she has offered, and her belief in me, has been a true source of joy and inspiration in my life. Words cannot express how grateful I am, and I only hope th at my support for her continues to be as meaningful as what she has given to me. I truly love you with all of my heart. I would especially like to express my appr eciation to my parents, John and Dianna Green, as well as Grandma, Gramps, Aunt Barb, and my extended family and friends whose love, support, and encouragement has been never ending. I could never have gotten this far without a ny of you. From the bottom of my heart, thank you. My thanks go out to my current and former group members: Russell DeVane, Christina Kasprzyk, Angela Perry, Ben Ro ney, Christine Neipert, Ashley Mullen, Katherine Forest, Chris Cioce, and especially Abe Stern and Jon Belof. They have been wonderful friends and colleagues. I would also like to thank my committee: Dr. Ra ndy Larsen, Dr. Milton Johnston, and Dr. Preston B. Moore for your time, wisd om, and honest advice. In addition, I thank Dr. Venkat Bhethanabotla for serving as my di ssertation committee chair. I have enjoyed working with all of you. Finally, I give thanks and appreciation to my advisor, Professor Brian Space, whose support and guidance have made this jo urney possible. I a ppreciate the amazing opportunities that you have offered me during my time working in your lab. Thank you.

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Table of Contents List of Figures iii Abstract v Chapter 1: Introduction 1 Chapter 2: Calculating the Molecular Volume of Azobenzene in Solution 5 2.1 Introduction ................................................................................................... 5 2.2 Materials and Methods .................................................................................. 9 2.3 Results and Discussion ................................................................................. 11 2.4 Conclusions ................................................................................................... 17 Chapter 3: SFG Spectroscopy of the Water/Vapor Interface 18 3.1 Introduction ................................................................................................... 19 3.2 Models and Methods ..................................................................................... 24 3.3 Results and Discussion ................................................................................. 37 3.4 Conclusions ................................................................................................... 48 Chapter 4: SFG Spectroscopy of the Ca rbon Tetrachloride/Water Interface 50 4.1 Introduction ................................................................................................... 50 4.2 Models and Methods ..................................................................................... 51 4.3 Results and Discussion ................................................................................. 53 4.4 Conclusions ................................................................................................... 58 Chapter 5: Third Order SFVS of the Silica/Water Interface 60 5.1 Introduction ................................................................................................... 60 5.2 Models and Methods ..................................................................................... 62 5.3 Results and Discussion ................................................................................. 67 5.4 Conclusions ................................................................................................... 72 Chapter 6: Computer Simulati on of Molecular Evolution 74 Chapter 7: Conclusion 79 List of References 81 i

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Appendices 91 Appendix A. Time Domain Expressi on for the Sum Frequency Response ........ 92 Appendix B. EVO Simulation Package .............................................................. 95 Appendix C. Site Rate Generator Package .........................................................111 Appendix D. AnchorSearch.cpp .........................................................................116 About the Author End Page ii

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List of Figures Figure 1.1: Schematic of an SFG Experiment ............................................................... 2 Figure 2.1: UVVis Spectra of Azobenzene ................................................................. 12 Figure 2.2: Equilibrium Solvated Structures of Azobenzene ........................................ 13 Figure 2.3: Volume Fluctuations of Azobenzene .......................................................... 15 Figure 2.4: Change in Molar Volume Betw een Conformations of Azobenzene .......... 16 Figure 3.1 SFG TCF Spectra Usin g Different Potentials ............................................. 32 Figure 3.2 SFG TCF Spectra Using Di fferent Polarizability Models .......................... 36 Figure 3.3 SFG TCF and INM Spectra for the Water/Vapo r Interface ........................ 38 Figure 3.4 SFG TCF Spectra for the O-H Stretching Region ...................................... 40 Figure 3.5 SFG TCF Spectra for the Intermolecular Region ....................................... 42 Figure 3.6 Water/Vapor Interface Snapshot ................................................................. 43 Figure 3.7 Probability Distributi on of the Direction Cosine ........................................ 44 Figure 3.8 Real and Imaginary Components of the Spectra ......................................... 46 Figure 3.9 Real and Imaginary Compone nts in the O-H Stretching Region ................ 47 Figure 4.1 SFG TCF Spectrum for the CCl4/Water Interface ...................................... 54 Figure 4.2 SFG TCF Spectrum for th e O-H Stretching Region of the CCl4/Water Interface ...................................................................................................... 55 Figure 4.3 Real and Imaginary Component s in the O-H Stretching Region of the CCl4/Water Interface ................................................................................... 56 Figure 4.4 CCl4/Water Interface Snapshot ................................................................... 57 iii

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Figure 5.1 Silica/Water Interface Snapshot .................................................................. 66 Figure 5.2 Third Order SFVS Spectrum for the Silica/Water Interface ....................... 68 Figure 5.3 Third Order SFVS Spectrum for the O-H Stretching Region of the Silica/Water Interface ................................................................................. 69 Figure 5.4 Second Order SFVS Spectrum fo r the O-H Stretching Region of the Silica/Water Interface ................................................................................. 70 Figure 5.5 Third Order SFVS Spectrum for the Silica/Water Interface with Increased Surface Charge ............................................................................................ 71 Figure 6.1 Interface for the EVO Simulation Package ................................................. 75 Figure 6.2 Interface for the Site Rate Package ............................................................. 76 Figure 6.3 Interface for the Anchor Search Package .................................................... 78 iv

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v Applications of Molecular Dynamics T echniques and Spectroscopic Theories to Aqueous Interfaces Anthony J. Green Abstract The primary goal of spectroscopy is to obtain molecularly detailed information about the system under study. Sum frequency ge neration (SFG) vibrational spectroscopy is a nonlinear optical technique that is hi ghly interface specific, and is therefore a powerful tool for understanding interfacial structure and dynamics. SFG is a second order, electronically nonresonant, polarization experiment and is consequently dipole forbidden in isotropic media such as a bulk liquid. Interfaces, however, serve to break the symmetry and produce a signal. Theoretical approximations to vibrational spectra of O-H stretching at aqueous interfaces are constructed using time correlation function (TCF) and instantaneous normal mode (IN M) methods. Detailed comparisons of theoretical models and spectra are made with those obtained e xperimentally in an effort to establish that our molecula r dynamics (MD) methods can re liably depict the system of interest. The computational re sults presented demonstrate th e potential of these methods to accurately describe fundamentally important systems on a molecular level.

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Chapter 1 Introduction Obtaining information about the structure and properties of liquid interfaces is an important, yet inherently difficult, field of study. Aqueous surfaces and interfaces are abundant in the environment, and essent ial to many chemical, biological, and atmospheric processes. Modern investigat ions into condensed phase structure and dynamics have successfully developed bett er detail on interfacial bonding. Such investigations have been conducted employing both theoretical [1-16] and experimental [17-44] methods to improve the understanding of the characteristics of liquid surfaces. Recent advances in vibrationa l spectroscopic methods specifically designed to observe liquid interfaces are being applied in experiments using nonlinear optical techniques such as Sum Frequency Generation (SFG) vibrational spectrosc opy. Collaboration between both experimental and theoretical methodologies as the field has evol ved has lead to the ability to describe complex c ondensed phase dynamics at the molecular level. However, more progress is clearly needed by both expe rimental and theoretical methods to fully understand surface dynamics. SFG is a vibrational spect roscopic method applied to evaluate nonlinear optical responses in the infrared frequency region. Th is technique allows for the measurement of vibrational spectra of interfaces by observing modes that are both infrared (IR) and 1

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Raman active [17]. Typically, signal is gene rated by combining visi ble laser pulses and tunable infrared laser pulses focused upon the interface to be studied, as shown in Figure 1.1. The resulting signal is a second-order response, which requires anisotropic media according to the dipole approximation [23]. In terfaces inherently disrupt the symmetry due to the lack of an inversion center which gi ves rise to the interface specificity needed. Figure 1.1: Schematic of an SFG Experiment. Schematic repr esentation of a typical SFG experiment at an oil/water interface. Structural and dynamical information about the interface can then be interpreted from the resonant enhancement to the re sponse that results when the infrared source frequency is coincident to molecular vibrations at th e interface [4,15,23,45]. The general theory of nonlinear spectroscopy is presented and disc ussed in chapter 3, specializing on second order processes. Obtaining molecularly detailed information contained in the vibrational spectroscopic signal critically depends on an accu rate interpretation of the spectra. The 2

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development of theoretical methodology can faci litate the interpretation of vibrational spectra obtained through experimental measurements, and is essential to understanding the properties of liquid interf aces at the molecular level. The work presented here includes the application of es tablished and newly developed simulation techniques that link the dynamics of interfaces directly to their experimental spectroscopy. Focusing on modeling experimental spectra serves the dua l purpose of interpreting the information inherent in vibrational lin eshapes and validating the mo lecular dynamics methods employed in simulating complex condensed phase systems. Specifically, these investigations consist of constructing molecular models of fundamentally and technologically important inte rfaces that are being investig ated experimentally using vibrational spectroscopy, including those c ontaining hydrophobic and charged surfaces. Using data obtained via molecular dynamics simulations, interfacial vibrational lineshapes can be calculated and interpre ted using both time correlation function and instantaneous normal mode theories. Chapter 2 outlines the molecular dynamics techniques used to calculate molecular volumes, and includes the results of the experi mental and theoretical determination of the molecular volume change associated with the cis-trans isomeriz ation of azobenzene. Chapter 3 discusses theoretical simulations of SFG spectra using a combined TCF and INM approach, with comparison to experi mental obtained data for the water/vapor interface. Results, including the identificat ion of novel species at the water/vapor interface, are presented. The work presented in chapter 4 expands the methods used in the previous chapter to probe the carbon tetrachloride/wa ter interface. The results obtained prompt 3

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further confidence in the methods as effec tive and accurate tools for interpreting SFG spectra. Chapter 5 discusses applications to charged interfaces, using an idealized silica/water interface as a precursory model. The development of a codebase for studying molecular evolution using computer simulation and phylogenetics is presented in chapter 6. Finally, chapter 7 presents conclusions a bout the work, and a brief discussion of potential future directions for theoreti cal studies of spectroscopy and molecular dynamics. 4

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Chapter 2 Calculating the Molecular Volume of Azobenzene in Solution In this chapter, the synergistically combined experimental and theoretical approaches to describe the molecula r volume change associated with the photoisomerization of aqueous trans -azobenzene to cis -azobenzene is presented. Although the cis isomer is sterically la rger, a volume contraction ( transcis ) of 4 mL/mol is observed by photoacoustic calorimetry in aqueous solution, as measured by Professor Randy Larsens research group at the University of South Florida. Theoretical methods predict the same volume contr action and have determined the origin to be due to electrostriction arising from the newly formed dipolar species. 2.1 Introduction One of the keys to understanding reaction mechanisms in both chemical and biological systems is having a complete kinetic and thermodynamic profile for a particular reaction. A thermodynamic parameter that is extremely useful in understanding structural aspects of a reaction, including the effects of solvent, is the molar volume [46]. Reactions, e.g. involving changes in bond lengths, bond cleavage/formation, metal spin-state transition, etc all result in characteristic changes in molar volume between the reactants and products. In biological systems, processes, such 5

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as protein conformational changes and proton or electron transfer reactions also produce significant volume changes. Change s in molar volumes accompanying a chemical/biochemical reaction are generally interpreted using generalized scaled particle theory in which the partial molar volume of a solute, V0, is expressed as: V0 = VM + VT + VI + T0RT; (2.1) where VM is the solvent excluded molar volume, VT is the volume that excludes solvent due to the thermal motions of the solute molecule, VI is a term that reflects changes in molar volume due to hydration and T0RT is the ideal term ( T0 is the isothermal compressibility of the solvent and R is the gas constant) [47,48]. The corresponding change in molar volume between two states of a solute molecule can be written as: V0 = VM + VT + VI (2.2) ( T0RT is a function of the solven t and is invariant between so lute conformations). An alternative description is to include VM and VT in a van der Waals term and classify the VI in terms of solvation (i ncluding electrostriction) V0 = Vintr + Vsolv, (2.3) where Vintr accounts for motion of the nuc lei (van der Waals term) and Vsolv includes volume changes due to changes in molecula r polarity, electrostric tion and multipolar 6

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interactions [48]. In pr actice, volume changes can be estimated using QSAR (quantitative structure activity relationships) for the Vintr term (provided crystal structures of the two states are available or relatively accurate models can be simulated) which involves either summing over the van der Waals volumes of the constituent atoms or by calculating the volume from the solven t accessible surface with a probe radius of zero. The corresponding Vsolv term can be estimated using the Drude-Nernst equation: VElect = B zi 2/ r (2.4) where r is the radius of the molecule and zi is the change in the overall charge on the molecule [49]. Although these methods have provided valuable mechanistic insights for a wide range of chemical and biochemical r eactions, they rely heavily on static X-ray structures. We have recently developed a method fo r calculating molecular volume changes using molecular dynamics (MD) methods [50]. In this method, the thermodynamic volume of a molecule is obtained by first performing isothermalisobaric MD on the solventsolute system for a specific length of time. The volume of the system is then calculated. The solute molecule is then plucked from the solvent box and a further simulation is performed. Once system equilibration is achieved the volume is again calculated. The difference be tween the two volumes is th e thermodynamic volume of the solute for the given solution. By performi ng these calculations on both the initial and final states of a molecule which undergoes a conformational change resulting in changes in dipole, etc., the thermodynamic V for the process can be obtained. This methodology 7

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accurately determined the molar volume of a flexible SPC water molecule and was used to probed the effects of aqueous solvation of a model methane molecule and fictitious methane derivatives with a perm anent dipole or full (+) and ( ) charges. These studies provided atomistic detail to electrostriction showing a much larger volume contraction for anions relative to cations in aqueous solution. Of the many experimental techniques capable of measuring changes in molar volume (e.g. dialatometry, equilibrium high pr essure optical and magnetic resonance and pressure dependent rate measurements ) only photothermal methods, including photoacoustic calorimetry (PAC) and photothe rmal beam deflection (PBD), can access changes in volumes on fast timescales dete cting differences occu rring on the order of tens of ns [51,52]. Photothermal experi mental methods can provide highly accurate molar volume changes but interpreting them is difficult because they reflect bulk thermodynamic volume changes. Extended ensemble molecular dynamics (EEMD) is capable of providing an atomis tically detailed description of molecular motions on time scales compatible with phot othermal techniques while tr acking any associated volume changes, and thus provides an ideal complement to such experiments [53]. In the present study, PAC is synergistically combined with EEMD to determine the molecular volume change associated with the photoisomerization of aqueous trans -azobenzene to cis azobenzene. Specifically, PAC is used to a ssess the magnitude of the resulting volume contraction and EEMD is used to theoretically determine the difference in molecular volumes while providing molecular reso lution to the origin of the change. 8

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2.2 Materials and Methods Cis and trans isomers of azobenzene were isolated as follows: ~5 mg of solid azobenzene (SigmaAldrich) was dissolved in ~2 mL of absolute ethanol. This solution was illuminated with a halogen lamp (150 W, a water filter was placed between the lamp and the sample) for ~30 min and then diluted with 5 volumes of deionized water and the solid trans azobenzene was filtered. All procedur es were performed under a red lamp to avoid photolysis. The trans samples of azobenzene for PAC were prepared by saturating a water solution with solid azobenzene (the trans isomer is only sparingly soluble in water). The absorbance at the excitation wavelength ( 355 nm) was 0.05 for both the azobenzene sample and the calor imetric reference compound (Fe(3+)tetrakis(4sulphonatophenyl)porphyrin). Samples were stirred during data acquisition and changed after every five laser pulses to ensure homogeneity since only the trans to cis transition was to be measured. Instrumentation for the photoacoustic measurements and subsequent data analysis has been described in detailed previously [54] Typically 10 laser pulses were averaged per trace. The PAC data were analyzed us ing the multiple temperature method in which sample and calorimetric reference acousti c traces are obtained as a function of temperature. The ratio of the amplitudes of the acoustic signals are then plotted versus 1/( / Cp) according to the following equation: (S/R) Eh = Eh = [ Q + ( Vcon/( / Cp))], (2.5) 9

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where is the quantum yield, Q is the heat released to the solvent, is the coefficient of thermal expansion of the solvent (K-1), Cp is the heat ca pacity (cal g-1 K-1), is the density (g mL-1) and Vcon represents conformational/electro striction contributions to the solution volume change. A plot of Eh versus Cp/ gives a straight line with a slope equal to Vcon and an intercept equal to the released heat ( Q ). Subtracting Q from Eh gives H for processes occurring faster than the time resolution of the instrument (<50 ns). The Q values for subsequent kinetic processes represent H for that step ( i.e. heat released). Contemporary molecular dynamics (MD) si mulations in the isothermalisobaric ensemble (NPT) were used to determine the molecular volumes of the cis -azobenzene and the trans -azobenzene using a code originally de veloped in the Center for Molecular Modeling at the University of Pennsylvania [ 55,56]. It has been pr eviously shown that the fluctuations of the volume coordinate in NPT MD are nearly Gaussian and thus the standard deviation of the volum e fluctuation is a useful meas ure of the uncertainty [57]. Care was taken in calculating the statistical inefficiency and, in turn, the correlation time of the fluctuations so that only uncorrelated measurements were averaged thus ensuring that each averaged configuration contained co mpletely uncorrelated information. It has been shown that this is equivalent to sa mpling more frequently and correcting for the correlation [50]. In order to obtain a reliable electrostati c potential surface, ab initio calculations were performed using the GAMESS package [ 58]. The 6-31G* basis was chosen for its over estimation of the (gas phase) charges (f it to reproduce the electrostatic potential surface); using over polarized charges accounts, on average, for condensed phase 10

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polarization effects. The charges were de rived by fitting to the electrostatic potential surface via the Connolly method us ing the GAMESS package [58]. A water-solvated trans -azobenzene, a water-solvated cis -azobenzene, and a neat water box were simulated classically in the NPT ensemble and parameterized via AMBER force field parameters augmented by the charges calculated in the ab initio simulations. The flexible simple point char ge (SPCF) water model was chosen for its computational efficiency and its reasonable reproduction of the condensed phase dipole, but our volume calculations depend on differen ces in volumes (or densities) that are accurately represented [50]. One hundred a nd eight water molecules were simulated in all three systems. Simulations were perfor med in the isothermalisobaric ensemble and allowed to run until the volume converged to an uncertainty of about 0.25 ml/mol. The calculated volume of the neat water NPT MD simulation served as the zero volume reference; the difference betw een the volume of the solvated azobenzene isomers and the volume of the neat water simulation were take n as the molecular volume of each isomer. The difference between the molecular volume of each azobenzene isomer was taken as the molecular volume change associated with the cis-trans isomerization of azobenzene, and compared to values obtained experimentally All calculations we re performed at The Research Computing Core Facility of The Un iversity of South Fl orida on an Intel Xeon cluster [50]. 2.3 Results and Discussion In the absence of light, both cisand trans -azobenzene can exist as metastable isomers in aqueous solution for hours. These isomers exhibit distinctive optical 11

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properties as shown in Figure 2.1. Upon illu mination, a fast photoisomerization occurs and a molecular volume change accompanies the conformational transition. The crystal structure of both isomers is known and trans -azobenzene is a planar molecule that lacks a dipole due to its symmetry, al though significant charge separati on is still present in the Figure 2.1: UV-Vis Spectra of Azobenzene. The figure presents the UV-Vis spectra of saturated solutions of both cis and trans -azobenzene in water. molecule (based on the ab initio calculations outlined below). Cis -azobenzene is a nonplanar molecule with a significant molecular dipole. A graphical representation of both isomers in their equilibrium geometry in bot h the presence and absence of the explicit solvent are presented in Figure 2.2. The coordinates were obtained by performing ab initio geometry optimizations at the 6-31G* le vel, and the resultin g structures were 12

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Figure 2.2: Equilibrium Solvated Structures of Azobenzene. In this figure, equilibrium solvated structures of cis (left) and planar trans (right) azobenzene are depicted. The top panels show the gas-phase structures for be tter visualization, and the bottom panels show the molecules solvated with 108 water molecu les. The atom types are represented as follows: red (O), white (H), green (C), blue (N). superimposable and visually indistinguishab le with those obtained from the crystal structures. Snapshots from EEMD simulations of the solvated isomers are also presented in Figure 2.2. The EEMD was performed in the isothermalisobaric (NPT) ensemble using reversible multiple time step integrat ion and the Amber ff99 force field and the simulations included 108 explicit water mo lecules. The partial charges on the azobenzene atoms were fit to the ab initio electrostatic potential surface using the Amber criteria and results in an equili brium gas phase dipole of 3.45 D for cis -azobenzene. 13

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The NPT EEMD exactly samples both the NPT ensemble and the associated fluctuations in the pressure and temperatur e variables giving desired average values here, ambient conditions are considered. The resulting dynamics are only slightly perturbed from true Newtonian motion and are sufficiently accurate to follow the time evolution of the system. During the simula tion the volume coordinate fluctuates over time and the average value is the system vol ume. Traces of the solvated azobenzene volume fluctuations compared with an equi valent amount of neat water are shown in Figure 2.3. Uncertainties in th e values are obtained from th e standard deviation of the nearly Gaussian fluctuations, and the coordina te is sampled at times that are sufficiently separated to be sta tistically uncorrelated [50,57]. For non-equilibrium simulations, e.g. following protein folding dynamics, this met hod of molecular volum e determination can resolve time dependent volumes and, given the inherent precision, identify intermediates that have lifetimes on the order of nanoseconds [50]. Here, equilibrium NPT EEMD simulations were performed for several nanoseconds and the volume of cis and trans -azobenzene were found to be 148.2 0.26 and 151.8 0.26 mL/mol, respectively, resultin g from 72 ns of EEMD. The volume difference between the isomers is 3.59 0.52 mL/mol with the cis isomer having the smaller solution volume. The absolute numbers compare favorably with the volumes obtained from the crystal structures and distin ct studies gave a range of values for the cis isomer of 149 mL/mol and for the trans isomer of 148 mL/mol [59,60]. While these numbers are not strictly comparable due to the different chemical environments involved, the agreement is striking. Also note, the trans isomer is slightly larger in the crystal state, in contrast to their aqueous behavior. 14

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Figure 2.3: Volume Fluctuations of Azobe nzene. Volume fluctuations for both cis (red) and trans (green) azobenzene, as well as neat water (blue) are presented over 100 ps of dynamics. Horizontal dashed lines represent the average volumes over the entire simulation. Molecular volumes are often discussed in the context of Van der Walls radii and how the overlapping spheres fill space, alt hough other effective methods to calculate molecular volumes exist [53]. While these steric interactions are a major contributor to molecular volumes, electrostatic interactions also play a large role, especially in dynamical volume changes. In fact, an earli er theoretical study demonstrated that an aqueous anion can have a negative volume in solution while the associated neutral species has a volume of 32 mL/mol; the pres ence of the anion causes the system to electrostrict to a volume smaller than the neat liquid [50]. To assess the role of electrostriction, the azobenzenes were simula ted without charges but otherwise using the 15

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same force field. The volumes of cis and trans -azobenzene were found to be 152.6 0.35 and 148.8 0.35 mL/mol, respectively, resulti ng from 40.5 ns of MD. The volume of the cis isomer is now larger, and this result is consistent with its larger crystal structure value where electrostriction is less important. This also implies that the cis isomers large dipole is responsible fo r an excess electrostr iction of 8 mL/mol in water, making the sterically larger cis isomer smaller in aqueous solution. A plot of the photoacoustic amplitude of th e sample (normalized to the reference) versus Cp / gives a line with a slope equal to Vcon associated with a change in conformation and an intercept equal to the heat evolved ( Q ) as shown in Figure 2.4. Since Q is the amount of heat released to the solvent associated w ith a reaction step, subtracting Q from Eh (the amount of energy absorb ed by the molecule) gives H / for the reaction. Photo-excitation of the trans form of azobenzene in water results in a volume change of 4 + 1 mL/mol (using a = 0.26) on a time scal e faster than can be Figure 2.4: Change in Molar Volume Between Conformations of Azobenzene. Plot of Equation 2.1 for the transition from trans to cis -azobenzene in water. 16

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resolved with the current instrumentation ( i.e. the PAC instrument employed for this study integrates all thermodynamic information occurring on time scales faster than ~50 ns) [55]. The observed volume contraction is similar to that observed in 80:20 ethanol:water, as well as for carboxyl-azobenzene in wa ter and are in excellent, quantitative agreement with th e NPT EEMD results where the cis has a smaller volume by 3.59 0.52 [61,62]. 2.4 Conclusions These studies demonstrate the potential of the combined approach accurate (time resolved) thermodynamic volume measur ements and NPT EEMD simulations to provide atomistic resolution of the origin of observed volume changes. While the methods are demonstrated here for a volume change between two equilibrium states the study demonstrates the ability of the combined approach to accurately describe the changes in shape and volume associated w ith intermediates al ong a reaction pathway. 17

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Chapter 3 SFG Spectroscopy of the Water/Vapor Interface An improved time correlation function (T CF) description of sum frequency generation (SFG) spectroscopy was developed and applied to theoretically describing the spectroscopy of the ambient water/vapor inte rface. A more general TCF expression than was published previously is presentedit is va lid over the entire vibrational spectrum for both the real and imaginary parts of the signa l. Computationally, earlier time correlation function approaches were lim ited to short correlation times that made signal processing challenging. Here, this limitation is overcome, and well-averaged spectra are presented for the three independent polarization condi tions that are possible for electronically nonresonant SFG. The theoretical spectra comp are quite favorably in shape and relative magnitude to extant experimental results in the O-H stretching regi on of water for all polarization geometries. The methodological improvements also allow the calculation of intermolecular SFG spectra. While the intermolecular spectrum of bulk water shows relatively little structure, the interfacial spec tra (for polarizations that are sensitive to dipole derivatives normal to the interfaceSSP and PPP) show a well-defined intermolecular mode at 875 cm1 that is comparable in intensity to the rest of the intermolecular structure, and has an intensity that is approximately one-sixth of the magnitude of the intense free O-H stretching pea k. Using instantaneous normal mode 18

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methods, the resonance is shown to be due to a wagging mode localiz ed on a single water molecule, almost parallel to the interface, with two hydrogens displaced normal to the interface, and the oxygen anchored in the interf ace. We have also uncovered the origin of another intermolecular mode at 95 cm1 for the SSP and PPP spectra, and at 220 cm1 for the SPS spectra. These resonances are due to hindered translati ons perpendicular to the interface for the SSP and PPP spectra, and translations parallel to the interface for the SPS spectra. Further, by examining the real and imaginary parts of the SFG signal, several resonances are shown to be due to a single spectroscopic species while the donor O-H region is shown to consist of th ree distinct speciesconsistent with an earlier experimental analysis. 3.1 Introduction Liquid water interfaces are ubiquitous and important in chemistry and the environment. Thus, with the advent of in terface specific nonlinear optical spectroscopies, such interfaces have been intensely studiedbo th theoretically [1-16] and experimentally [17-44]. Sum frequency generation (SFG) spectroscopy is a powerful experimental method for probing the structure and dynamics of interfaces. SFG spectroscopy is one of several experimental methods that measur e a second-order polarization, and the more common electronically nonresonant experiment is considered here. SFG spectroscopy is dipole forbidden in isotropic mediasuch as liquids. Contributions from bulk-allowed quadrapolar effects have been demonstrated to be negligib le in some cases [63,64], but can be included if necessary [65]. Interf aces serve to break the isotropic symmetry, and produce a dipolar second-order signal. The SFG experiment employs both a visible and 19

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infrared laser field overlapping in time and sp ace at the interface, and can be performed in the time or frequency domain [23,66]. In the absence of any vibrational resonance at the instantaneous infrared laser frequency, a structureless signal due to the static hyperpolarizability of the in terface is obtaine d [5,20,26]. When the infrared laser frequency corresponds to a vibration at the interface, a resonant lineshape is obtained with a characteristic shape that reflects bot h the structural and dynamical environment at the interface [2,67,68]. In this chapter, classical molecular dynamics (MD) methods are used to model the dynamics of the water/vapor interface. Two complementary theoretical approaches quantum corrected time correlation functi on (TCF) and instantaneous normal mode (INM) methodsuse the configurations genera ted by MD as input to describe the SFG spectrum of the interface, and to ascertain the molecular origin of the SFG signal; both INM and TCF methods rely on a suitable spectro scopic (dipole and polar izability) model. This dual approach was demonstrated to be highly useful in understanding condensed phase spectroscopy of water, other liqui ds, and interfacesclassical mechanics, especially in the context of quantum-corrected TCFs, ha s proven to be surprisingly effective in modeling intramol ecular vibrational sp ectroscopy [1,2]. In particular, TCF methods have provided a quantitative descri ption of the O-H stretching lineshape in ambient liquid water, and INM methods have served to identify the molecular motions that result in the observed signal; these co mplementary techniques are equally effective for modeling water interfaci al spectroscopy [1-6,69-75]. An INM approximation to SFG spectro scopy is quantum mechanical by construction, but offers a limited dynamical de scription. As a result, in bulk water (and 20

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other liquid state intramolecula r lineshapes), INM intramolecular resonances are broader than their TCF counterparts, but have the same central frequency and integrated intensity. This observation suggests the intramolecu lar INM spectra represent an underlying spectral density that is dynamically motionally narrowed in the actual lineshape [67]. This is also found to be the ca se here for SFG spectra in al l polarization conditions. This result contrasts with a previous report by Perry et al [1], and evidence from the literature [5,6]. Previous TCF and INM calculations of the (SSP polarization) SFG O-H stretching spectrum of the water/vapor interface were very noisy, and suggested the spectra had equal breadththus, suggesting motional narro wing effects were not apparent in the spectra. The success of an approximate, nondynamical, frequency domain technique [6], and the similarity of the spect ra to those obtained using TCF methods [1,5], appeared to be further evidence of spectra that could be described in the i nhomogeneously broadened limit [76]. That method [6], however, contains an empirically adjustable line width that effectively accounts for some motional narrowin g making it difficult to draw conclusions. Because of methodological advances, it is now possible to calculate well-averaged TCF and INM spectra, and they unambiguously de monstrate SFG O-H stretching lineshapes (at least at the water/vapor interface) are significantly motionally narrowed to a degree reminiscent of the bulk [74,77]. This observation suggests dynamical motiona l narrowing effects are important at interfaces, and the dynamics are best described as intermediate between the fast and slow modulation limits of motional narrowing. In the slow modulation inhomogeneously broadened limit, all frequency fluctuations of the oscillator are represented in the lineshape [67,76,77]. A recent study in the Shen group also suggested motional 21

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averaging effects may well be si gnificant in the SPS geometr y, and, in that case, the free O-H stretching peak is greatly diminished. Although that study did not address motional narrowing, the presence of mo tional averaging suggests moti onal narrowing is important because it is due to fast reorientational moti ons within the vibrational relaxation time for the mode that would also be expect ed to result in motional narrowing. In order to obtain better TC F results, long time (cross) correlations between the system dipole and polarizability need to be followed. Because molecular simulations of interfaces in Cartesian space necessarily pr oduce two interfaces, simulation times were limited to the molecular diffusion time between interfaces so molecules could not contribute to the signal at both interfaces durin g one MD run [1]. This leads to TCFs without long time decays that are difficult to Fourier transform accurately. In this work, a weak restraining potential is added that conf ines the molecules over time to the half of the simulation box they start in (in the di mension normal to the interface) without significantly perturbing the (rel evant short time) dynamics a nd average structure of the liquid that contributes to the interfacial spectroscopy; even though the molecular diffusion constant (normal to the interface) is changed, the molecule is only contributing to the spectrum while resident at the interface, and is free of any significant external potential. This modification permits the calculation of TCFs out to arbitrarily long times resulting in sharp spectra that include inte rmolecular spectral lineshapes. Surprisingly, a welldefined intermolecular mode was found to be prominent in the spectrum [2]. It is centered at 875 cm1, and is comparable in (integrate d) intensity to the rest of the intermolecular lineshapethe lineshape also has an intensity that is approximately one22

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sixth of the magnitude of the intense free O-H stretching peak for spectra taken in polarization geometries that are sensitive to dipole derivatives normal to the interface (SSP and PPP). Using instantaneous normal mode methods, the resonance is shown to be due to a wagging mode localized on individu al water molecules. Water molecules contributing to this resonance are at a slight angle to the inte rface with their oxygen atoms anchored in the interface, and the hydrogen atoms wagging nearly normal to the interface. The presence of another population, aside from the free O-H stretch, of interfacial molecules was recently proposed via indirect evidence [8,43,44], and that hypothesis is strongly supported by this work. Here, we have directly observed a spectroscopically distinct specie s, and clearly identified the vibrational mode responsible for the lineshape. Thus, experimental setups that permit taking spectra at relatively long wavelengths could probe this mode as a comp lement to the information contained in the free and donor O-H stretching modes [78-83]. At lower frequencies, well defined hindered translational modes are found both para llel and perpendicular to the interface. The perpendicular modes are prominent in the polarization conditions sensitive to dipolar changes normal to the interface (SPP and PPP) while the parall el modes are more pronounced in the SPS geometry which is se nsitive to motions along the interface. Further, some of the time domain expr essions given earlier for SFG spectroscopy were only correct for the modulus of the SFG signal (but not for the real and imaginary portions) [1,7], or were not entirely general (and only va lid at high intramolecular frequencies [5,65]. Note, some of these wo rks also equated the (complex) quantum TCF with the (real) classical TCF, further comp licating matters [5,65]. The correct general and exact expressions are given below and include both the res onant and nonresonant 23

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contributions. The previous expressions all give acceptable lineshapes (at high frequencies) for the modulus of the secondorder signal. They are not correct for calculating the amplitude of the signal that can be detected in a heterodyne experiment [58], or by taking advantage of phase interference effects [ 17]. Here, it is shown by carefully examining the real and imaginary parts of the SFG signa l, individual mode contributions to an observed lineshape can be identified. Using this approach, the free O-H and the newly discovered modes were id entified as individual spectroscopic species (one type of oscillator at the interface), and the donor O-H region consists of three distinct species. This last conclusion agrees with results from a careful deconvolution of O-H stretching signal in an ear lier experimental work that also found three specieseach with approximately the same central frequency [20,85]. Thus, as a prelude to more complex interfaces, this joint TCF/INM approach is applied to the water/vapor interface producing good agreement with the shape and relative amplitudes of SFG measurements for all independent polarization conditions [68]. The theoreti cal expression in terms of a TCF for the SFG signal is also presen tedincluding corrections from expressions published previously [1,5,7] in the next se ction and Appendix A. The MD, dipole, many body polarization methods, and associated pa rameters are also summarized in the following section. The theoretical results, and their comparison to experiment, are discussed later in the chapter as well. 3.2 Models and Methods The SFG signal consists of nonresonant (due to the static hyperpolarizability) and resonant contributions that are important when the infrared laser fr equency distribution is 24

PAGE 33

resonant with a vibrational transition at the interface. The signal intensity, is proportional to the square of their sum: and directly measures the modulus squared in the typical homodyne detected effectively monochromatic frequency domain experiment. The superscripts deno te the resonant and nonresonant contributions respectively, and is the susceptibility tensor. The proportionality constants include a factor of 2 and Fresnel factors [86]. The vibrational information is contained in the resonant signal, and the nonresonant was found to be a (negative) constant in the O-H stretching region [5,20]; this can still lead to the nonresonant contribution changing the fre quency-dependent intensities through cross terms in the squared modulus signal. Thro ugh isotopic substitu tion, the nonresonant contribution can be measured independently, and this permits the deconvolution of the full signal to extract Res(); this deconvolution was done for the SSP polarization geometry at the water/vapor interface [20]. 2 Re Re 2)2(|)()(|)|(|sN s SFGI )()2(SFGThe second-order response is given theo retically by a combination of resonant and nonresonant terms [5,25,84,87]. The resonant terms can be grouped to give a simple expression in terms of the systems polarizability and dipole. A derivation including the resonant and nonresonant terms fr om perturbation theory is given in Appendix A [25,87]. Thus Res() is given by [7,84] )}(],{[ )(0t Tredt ijki ti R (3.1) 25

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In Equation 3.1, = e H/Q for a system with Hamiltonian H and partition function Q at reciprocal temperature = 1/kT, and k is Boltzmanns constant. The system dipole is represented by and is the polarizability tensor of the dipolewhere the subscripts represent the vector and tensor components of interest, respectively. The operator evaluated at time t is the Heisenberg representation of the operator jk(t) = eiHt/ jk e-iHt/ ; Tr represents the trace of the operato rs. It is convenient to proceed by rewriting the FourierLaplace transform in Equation 3.1 as the F ourier transform of a correlation function that can then be interp reted in the classical limit, and quantum corrected. Evaluating the commutator in Equation 3.1 gives Tr{[ ,i] jk(t)} = C(t) C*(t) = 2i CI(t), (3.2) where C(t) = i jk(t) = CR(t) + iCI(t). In Equation 3.2, the superscript aste risk is the complex conjugate, and the subscripts denote the real and imaginary parts of C(t)both of which are themselves real functions. The angle brackets are the trace of the operators divided by the partition function in the standard notation [88]. In the classical limit, CR(t) becomes the classical cross-correlation function of the system dipole and polarizability tensor elements, i.e., lim 0CR(t) = CCl(t) = i jk(t). Since only classical TCFs can be calculated using classical MD and TCF theory, the goal is to write the response function entirely in terms of the (quantum-corre cted) classical TCF, CCl(t). 26

PAGE 35

To proceed, the imaginary part of the one time correlation function is related in frequency space exactly to the real part: CI( ) = tanh(/2)CR( ); where the subscripts denote the Fourier transform of the real a nd imaginary parts of the complex function C(t) which is a real function of frequency, i.e., )()())()(( 2 1 )( I R I R tiCCtiCtCedt C (3.3) Using the result obtained in Equation 3. 2 for the trace in Equation 3.1 gives 0 Re)( 2 )( tCedtI ti s, (3.4) )'()2/'tanh(')(' R ti IC edtC Equation 3.4 demonstrates the SFG expe riment probes the imaginary part of C(t) Note, CI(t) is written in a form that can be calculated using the real part of the correlation functionwhich is obtainable from classical MD. Due to causality, the FourierLaplace transform gives a real and imaginary part in Equation 3.4 as the cosine and sine transform of CI(t) respectively. Equation 3.4 can be simplified by changing the order of integrationperforming the frequency domain integral first. Defining the real and imaginary parts of : ) ()()(Re Re Res I s R si 27

PAGE 36

dttCt CI R s I)()sin( 2 )()2/tanh( 2 )(0 Re (3.5) 0 Re)()cos( 2 ' )'()2/tanh( 2 )( dttCt d CI R s R (3.6) To obtain Equations 3.5 and 3.6, the identity )(1 0 idteti was used, where. .designates the principal part [89]. Due to technical constraints in producing intense tunable infrared laser light, the focus of SFG experiments is curren tly on high frequency spectra where kT and classical mechanics is clearly invalid. Building on our previous work, the classical correlation function result, that is amenable to calculation using MD and TCF methods, is quantum corrected using a harmonic correction factor: CR( ) = CCl( ) [( /2) coth ( /2)] [1,90]. This correction factor is exact in relating the real part of the classical harmonic coordinate correlati on function to its quantum mechanical counterpart. Here, we are using it to correct functions, the dipole an d polarizability, that contain higher orders of the coordinates, and exact corrections for harmonic systems of this type are still possible, but not need edthe linear dipole and Placzek approximation are adequate [90]. Using this result, the TC F approximation to the resonant part of the SFG spectrum, Res, takes the form )()(Cl TCF IC (3.7) ' ')'( )( d CCl TCF R, (3.8) 28

PAGE 37

CCl(t) = i(0) jk(t) (3.9) In Equation 3.9, the angle brackets represent a classical TCF that can be computed using MD, and a suitable spectroscopi c model [91]. Finally, Equations 3.7 and 3.8 give the TCF signal in a form amenable to classical simulation. Note, while it is easier to evaluate I( ) using Equation 3.7, R( ) is more easily computed by doing the cosine integral as in Equation 3.6. Considering the three possible independent polarization conditions (SSP, PPP, and SPS) for the TCF in Equation 3.9, the first index in the polarization designation corresponds to th e last index in the TCF. For example, the SSP and PPP polarization conditions probe dipolar motions normal to the interface, and the SPS case is sensitive to dipolar changes parallel to the interface. Note, the PPP condition is sensitive to moti ons both parallel and perpendicu lar to the interface [68]. Further, the (SSP and PPP)/( SPS) probe diagonal/off-diagonal polarizability matrix elements, respectively. A similar TCF approach was adopted earlier by others [5,7] and Perry et al [1] for modeling the SFG spectrum of both solid [7] and liquid interfaces [1,5]note, quantum corrections were not included in the works by the other groups. The earlier papers, and previous work, did not give the exact expression for the real and imaginary parts of the SFG signal. Two of the papers [1,7], started with the time domain expression for the second-order response function [84], one improperly evaluated a contour integral which violated causality. This effectively eliminated the r eal frequency contribution, and doubled the imaginary frequency part of the susceptibility. The other work [5] used frequency domain perturbation theory [25,87] and divided the terms into resonant and 29

PAGE 38

nonresonant contributions then recast the resona nt contribution as a TCF. This led to a TCF expression that replaced CI(t) with the full TCF, C(t), in Equation 3.3 (to within a constant factor). Note, at high frequency, the tanh factor is almost unity, and their expression is a correct limiting expressionby making the rotating wave approximation, the expression is only correct at high frequenc ies [84,93]. In that work, if two of their nonresonant terms are included in the resonant contribution, then the exact Equation 3.4 is obtainedthis result is demonstrated in Appendix A. At lower frequencies, the two expressions are quite different, and the tanh factor produces a time derivative of the correlation function in the time domain. To construct an INM approximation to Equation 3.1, it is sufficient to evaluate the trace in Equation 3.1 for a harmonic system. It is convenient to invoke both the Placzek and linear dipole approximation to evaluate the resulting matrix elementsalthough higher-order contributions can be included, and simple analytic expressions result for these contributions. An equivale nt approach is to evaluate CCl(t) for classical harmonic oscillators, and quantum correct the resulti ng expression using the harmonic correction factor, given above, to relate CCl(t) and CR(t) : )()/)(/()(2 l ljkli ClQ Q kT C (3.10) In Equation 3.10, l is the frequency of mode Ql, and the angle brackets represent averaging over classical confi gurations of the system gene rated. This expression can then be back transformed into the time doma in, and used in Equations 3.7 and 3.8 in place of the classical TCF to obtain an INM approximation to the spectroscopy. Below, 30

PAGE 39

it will be demonstrated that the TCF a pproach, which does not invoke the Placzek and linear dipole approximation (except implicitly in quantum correcting the results), gives results in close agreement with the INM results, and Equation 3.10 is therefore sufficient. MD simulations were performed using a code developed at the Center for Molecular Modeling at the University of Penn sylvania, and uses reversible integration and extended system techniques [94]. Micr ocanonical MD simulations were performed on ambient H2O with a density of 1.0 g/cm3, and an average temperature of 298 K. To create an interface, a cubic simulation box of equilibrated liquid water was extended (doubled) along the z axis, and the system wa s allowed to equilibrate, creating two water/vapor interfaces. The in terfaces were sufficiently far apart so as they did not interact strongly, and Ewald summation was included in three dimensions [11]. The density profile of the system was monitored to verify equilibration [11]. In all cases, the results were tested, and found to be syst em-size independent. Most results were generated from 216 molecule simulations and smaller system sizes down to 64 molecules were tried, and di d not alter the results [2]. As in previous work [1,2,72,74], MD simu lations were conducted using a flexible simple point charge (SPC/F) model that included a harmonic bending potential, linear cross terms and Morse OH stretching potentials, V(r) = De(1er)2 [74]. The Morse O-H stretching potential used here was slight ly softer than previous work; here, the value is 2.50185 1 instead of 2.566 1 [1,74]. For a Morse potential, the force constant, k can be approximated as k = 2 De 2. Assuming a harmonic os cillator with frequency mk / this implies the ratio 1 : 2 is proportional to the ratio 1 : 2. Therefore, a 2.5% change in the exponential Morse parame ter implies a 2.5% sh ift in the spectral 31

PAGE 40

frequencies, and this behavior is demonstrated in Figure 3.1. This analysis assumes the relevant coordinates are simple one-dimensi onal O-H stretching modes. If several distinctly different types of modes were present, a change in the shape of the broad O-H stretching signal would be expected. This is additional evidence that interfacial normal modes are well approximated as simple O-H stretches [1,2,6]. Figure 3.1 highlights the spectral change s resulting from using a softer Morse potential. The slightly softer potential doe s not alter the intermolecular region of the spectra as would be expected. The intermolecular portion of the spectrum has polarizability and dipole derivati ves (changes) that are due prim arily to reorientation. Figure 3.1: SFG TCF Spectra Using Different Potentials. SFG SSP TCF spectra for the water/vapor interface highlighting the spectral changes in the use of two different Morse potentials the original Mors e potential (dashed blue line), and a softer Morse potential (solid green line). The softer potential results in a shift of approximately 100 cm1 in the O-H stretching spectrum. 32

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These changes then depend on the polarizability tensor and the dipoles themselves, and not their derivatives. On the other hand, the intramolecular region of the spectra is simply shifted to the redthis point will be returned to when discussing the modal composition of the broad O-H stretching lineshape. This change resulted in the free O-H stretching frequency in bette r agreement with experimental values even though the Morse potential change is almost imperceptible to the naked eye. This implies the center of the lineshape is very sensitive to the local fr equency along the Morse potential as the O-H stretching motion, perturbed by hydrogen bo nding in the liquid, explores the highly anharmonic potential surface. In performing the MD, partial point charge s were placed on the atoms that were chosen to reproduce the condensed phase dipo le moment. At the water/vapor interface, the true water dipole falls from its condensed phase value, about 2. 9 Debeye, to that in the gas phase, 1.8 Debeye, over a distance of only a few molecular layers [95]. It would seem polarizable dynamics would be esse ntial to model the dynamics of aqueous interfaces, but the use of nonpolarizable MD se ems to adequately represent the structure of the water/vapor interface. A previous work using a polarizable model in this context is consistent with this observation [5]. Evaluating the TCF in Equations 3.7.9 presents a problem for interfacial systems. The interface was constructed using the standard MD geometry with vacuum/vapor above and below the water [6 ,10]. Unfortunately this produces two interfaces with average net di poles in opposite directions. Calculating the SFG spectrum of the entire system would lead to partial cancellation of the SFG si gnal, and meaningless results. Another problem arises in that mol ecules at one interface can diffuse to the other 33

PAGE 42

interface over time. In this case, simulati on times are limited to the molecular diffusion time between interfaces, so that molecules can not contribute to signal at both interfaces during one MD run. This leads to TCFs without long time decays that are difficult to Fourier transform accurately [1,5]. In order to obtain better TC F results, long time (cross) correlations between the system dipole and polarizability need to be followed. A weak (laterally isotropic) restraining potential was added, effectively confining the molecules over time to the half of the simulation box they start in (in th e dimension normal to the interface) without significantly perturbing the relevant short time dynamics; even though the molecular diffusion constant (normal to the interface) is changed, the molecule is only contributing to the spectrum while resident at the interface, and is free of any significant external potential. This modification permits the calcu lation of TCFs out to arbitrarily long times, resulting in sharp spectra that include intermol ecular spectral linesha pes. The restraining potential is of the form V = ( / r )9 with =2.3 K, =2.474 and r = 0 is at the center of the box. The restraining potential be comes negligible near the interface, and is only significant within 2.0 of the box center. The interfacial density profile was unchangeddemonstrating the restraining pote ntial used did not perturb the average structure of the liquid that contribut es to the interfacial spectroscopy. The MD was performed wit hout explicit polarization fo rces; when the SFG TCF or INM spectrum are calculated, polarizabil ity is included in the calculationsover 3 million 3 fs time steps were included in calcu lating the MD and TCFs. The model atomic polarizability approximation (PAPA) pol arizability model [ 96-98] with point polarizabilities on the atoms (O = 1.1482 3, H = 0.3304 3) [99]. The permanent 34

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dipoles were calculated based on ab initio data as described in previous work [1,5]. The SFG signal is sensitive to both dipole and polar izability derivatives. PAPA polarizability models naturally incorporate pa rameters that determine the polarizability derivatives. To implement this, it is sufficient to make the point polarizabilities on the atomic centers (O-H) bond-length dependent [ 96-98]. The point polarizabil ities then change as (r) = 0(r) + r where r is the displacement from the equilibrium bond length. The parameters for hydrogen and oxygen (O = 2.7 2, H = -1.06 2) are somewhat different than in previous models, but stil l give reasonable values for the gas phase Raman and IR transition moments [1]. Figur e 3.2(c) highlights the differences between the previous and current model for the SFG SSP TCF spectra (showing the same SFG SSP spectrum presented in Figure 3.1). It is necessary, but not sufficient, to simply match the gas phase spectroscopic data, and s uggests that interfacial molecules explore geometries different from both the gas phase and the bulk (whe re the earlier polarizability model worked very well) [72,74]. Fitting to ab initio data for these interfacial geometries is clearly desirable, and is being pursued [5]. Further, point atomic polarizability models, such as the one used here [1,72,96,99,100], offe r flexible and transferable parameters for both neat mixtures and liquids. They also offer a natura l description of the induced dipole derivatives, and the ability to fit polar izability derivatives [98]. However, to produce an accurate description of interfacial polarizability derivatives, it may be necessary to make the poin t polarizabilities depend on bond angles, and not simply bond lengths as was done in this work [1]. It is interesting to note that the ne w model captures the free O-H mode more accurately without significantly perturbing the intermolecular region of the spectra 35

PAGE 44

intramolecular spectra are sensitive to dipol e and polarizability derivatives that do not significantly change the magn itude of the dynamically more important dipole and polarizability. (Note, the small differences in the intermolecular spectrum are likely due to the relatively poor averaging that was done in calculating the spectrum using the previous model. In this case only one-fift eenth of the number of configurations were included in the calculation, and the SFG TCFs were slow to converge [1,65].) Thus, even relatively small changes to these derivatives can greatly effect the spectroscopic observable, without changing the essential physics of the problem e.g. the identity of the relevant modes and motions. Figure 3.2: SFG TCF Spectra Using Different Polarizability Models. The (a) IR TCF spectra for liquid water, the (b) isotropic Raman TCF spectra for liquid water, and the (c) SFG SSP TCF spectra for the water/vapor inte rface highlighting the sp ectral changes in the use of two polarizability models prev ious model (dashed blue line) and current model (solid green line). 36

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Figure 3.2 also presents the (a) infrar ed and (b) isotropic Raman TCF spectra (relevant to the SSP polarization condition becau se it probes diagonal elements of the polarizability matrix) for liquid water using both models. Again, only the intramolecular region of the spectra ch anged. For the O-H stretching re gion, increased asymmetry in the lineshape is apparent for the new model with a shoulder on the blue side. This is consistent with previous work that identified this shoulder to be due to instances in which a hydrogen does not form a hydrogen bond in the bulk [101,102]; this would be analogous to the free O-H stretch found in inte rfacial spectra. The new polarizability model does a better job at highlighting this non-hydrogen bonded frequency distribution for liquid water, and, consequently, allows for more accurate interf acial spectra. The figure also clearly demonstrates the power of calculating spectrosc opic observables to analyze condensed phase and interfacial structur e. Interestingly, th e shoulder on the blue side of the bulk Raman and IR spectrum is at the same central frequency as the free O-H mode at the interfacestrongly suggesting the presence of free, non-hydrogen bonded, O-H modes in bulk water. 3.3 Discussion Figure 3.3 displays the theoretical SFG SSP spectra for the entire water vibrational spectrum derived from both TCF and INM methods. Both the TCF and INM results are in absolute units, and no parameters were adjusted in displaying the data. The INM and TCF spectra were found to integrat e to the same value over the entire 0 cm1 range, and separately over th e O-H stretching region (2000 cm1) for all polarization conditions (the others are not shown). This be havior is strong evidence for 37

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Figure 3.3: SFG TCF and INM Spectra for th e Water/Vapor Interface. SFG SSP spectra for the water/vapor interface for the entire water vibrational spectrum using TCF (solid green line) method and INM (d ashed blue line) method. the interpretation of the INM lineshape as an underlying spectral density that is motionally narrowed in the observed spec trum [67]. INM approximations to spectroscopy offer only a limited dynamical desc ription, and correspond to an underlying spectral density that is typically broader th an the observed lineshape when considering intramolecular modes. As an example, in bulk water (and other liquid state intramolecular lineshapes) INM intramolecular resonances were found to be broader than their TCF counterparts, but with the same central frequency and integrated intensity. The TCF and INM spectra in Figure 3.3 unambi guously demonstrate SFG O-H stretching lineshapes at the water/vapor interface are significantly motionally narrowed to a degree reminiscent of the bulk [74,77] This result also suggests SFG spectra are sensitive to 38

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both structure and dynamics. The INM spectru m clearly exhibits the same resonances, but is broader. This implies that the obs erved lineshapes are motionally narrowed, and dynamical contributions to SFG signals are important [68]. Figure 3.4 presents TCF derived theoretical descriptions of the SFG spectra in the O-H stretching region for the water/vapor in terface. The three possible independent polarization conditions (SSP, PPP, and SP S), in the electron ically nonresonant experiment are displayed. Th e first two indices can be inte rpreted as the element of the system polarizability tensor, and the second index as the element of the system dipole that is being probed. In the data, for all polarizations, we have included the SSP nonresonant contribution (this is only strictly correct for the SSP polarization condition, and serves as an estimate in the other cases), N Res( ), which is a small negative constant [5,85], and the full signal is given by In order to account for the Fresnel coefficients that modi fy the experimental intensities, we have adjusted the relative intensities of our theo retical spectra so they can be more easily compared with experimental results [68]. 2 Re Re 2)2(|)()(|)|(|sN s SFG The spectrum in the SSP geometry that co rrelates the dipole moment component normal to the interface with dia gonal polarizability matrix elements in the plane of the interface [ e.g., z(0) xx( t ) with the z axis taken as the surface normal direction] leads to the most intense spectrum due to a relativ ely sizable, and changing, net normal dipole moment at the interface, and the relatively large diagonal polarizability elements; water has a nearly diagonal polarizab ility matrix with nearly eq ual elements in both the gas phase and bulk. (Note, the PPP polarization condition is sensitive to a combination of all allowed susceptibility te nsor elements in contrast to SSP and SPS that only probe a single 39

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Figure 3.4: SFG TCF Spectra for the O-H St retching Region. SFG TCF spectra for the water/vapor interface in the O-H stretching re gion for three polarizations: SSP (solid green line), PPP (dashed blue li ne), and SPS (dotted red line). tensor element [92]). Previously, it has been shown that the agreement between the TCF and experimental spectrum, including the relative intensities of the different polarization conditions, was excellent, and within the stat istical error over most of the frequency range [2]. Thus, the essential features of the spectrum, and its polarization dependence, are captured very well by the TCF theory with the caveat that absolute intensities of the intramolecular modes are quite sensitive to the choice of polarizability parameters. The polarization dependence of the signal is demonstrated in Figure 3.4. For polarizations that are sensitive to dipol e derivatives normal to the interfaceSSP and PPPthe signal has an intense lineshape. In contrast, for the SPS geometry, which is sensitive to dipole derivatives parallel to the interface, only a hint of a signal is found. 40

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The SPS polarization condition also probes small off-diagonal polarizability matrix elements. These results also suggest by eval uation of the polarization dependence of the SFG spectra, given a knowledge of the expected nature of the polarizability and dipole derivatives, allows interfacial molecular ge ometries to be inferred via the spectra [17,92,103]. While the intermolecular spectrum of bulk water shows little structure, the interfacial spectra are complex as shown in Figure 3.5. The figure highlights the intermolecular SFG TCF spectra for the thr ee independent polarization conditions (SSP, PPP, and SPS). The polarizations that are sensitive to dipole derivatives normal to the interface, SSP and PPP, show a well-d efined intermolecular mode at 875 cm1 that is comparable in intensity to the rest of the intermolecular structure and approximately onesixth the intensity of the in tense free O-H stretching peak [2]. Using instantaneous normal mode methods (looking at the nature of the INMs in the same spectral region), the resonance is shown to be due to a wagging mo de localized on a single water molecule, at a slight angle to the interface, with two hydrogens vibrating/librating normal to the interface, and the oxygen anchored in the interface [2]. Th e hydrogens, pointing into the vapor phase, are hydrogen bonded to an oxygen atom at the interface. The SSP and PPP also show an intense intermolecular mode at 95 cm1. Using instantaneous normal mode methods, the resonance is found to be due to translations perpendicular to the interface. The SPS spectra which are sensitive to dipole derivatives parallel to the interface, show an intermolecular mode at 220 cm1. This mode is a result of translations parallel to the interface. The importance of polarization sensitivity in SFG 41

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Figure 3.5: SFG TCF Spectra for the Interm olecular Region. SFG TCF spectra for the water/vapor interface in the intermolecular region for thr ee polarizations: SSP (solid green line), PPP (dashed blue li ne), and SPS (dotted red line). experiments is, thus, highlighted. Further, we have observe d spectroscopically distinct species, and clearly identified the vibrationa l modes responsible for the lineshape. Hence, experimental setups that permit ta king spectra at rela tively long wavelengths could probe these modes as a compliment to the information contained in the free and donor O-H stretching modes [52,54]. These thr ee distinct populations of water molecules at the interface were previously undescribedother works have inferred the existence of something like the wagging mode [8,43,44]. Th is might be considered surprising given the large numbers of MD simulations of the water/vapor interface that have been performed previously. This observation highl ights the power of cal culating spectroscopic observables in assessing interfacial structure and dynamics. Not only can the results be 42

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directly compared with experiment, thus validating the MD model, the spectroscopic calculation serves as a filter of the dynamics extracting out the identity of collective coordinates with well-defined freque ncies that persist at the interface. Figure 3.6 highlights the vibrational modes from the intermolecular and intramolecular region of the sp ectra. A typical free O-H mo de, shown in blue, produces the high frequency feature at 3700 cm1. It is clear the oxygen atom is anchored in the interface, and the O-H is oscillating freely above the interface. The wagging mode giving rise to the spectral feature at 875 cm1 is displayed in green at the opposite interface. Here, the oxygen atom is anchored in the in terface, and the two hydrogens are vibrating into the vapor phase. A representative perpe ndicular translational m ode (with lineshape Figure 3.6: Water/Vapor Interface Snapshot A snapshot of a water/vapor interface containing 216 water mol ecules featuring INMs from diffe rent regions of the spectra. The water molecule shown in blue is representative of a free O-H mode at 3694 cm1. The water molecule shown in green is re presentative of a wagging motion at 858 cm1. The water molecule shown in yellow highli ghts a translation perpendicular to the interface at 46 cm1. The water molecule shown in black highlights a translation parallel to the interface at 197 cm1. 43

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centered at 95 cm1) is shown in yellow, and the roughly parallel transla tional mode (with lineshape centered at 220 cm1) is shown in black. These results demonstrate how the INM approach does not require a priori assumptions about the nature of interfacial modes, but does reveal their physical char acteristics, and how different molecular motions contribute to the spectrum. In fu ture work, a quantification of the relative populations of these interfacial speci es is planned via this approach. Figure 3.7 displays the dist ribution of the direction cosine from the surface normal of O-H vectors pointing into the vapor. This result compares well with previous theoretical data [6]. An enhancem ent in probability is seen at cos 1. We also find that approximately 20% of surface water molecule s have a free O-H bond pointing out of the liquid and into the vapor, which is consistent with previous theoretical work [6,10] This analysis also points out that it is necessary to talk of broad distributions of angles at the water/vapor interfaces, and that relatively less can be learned from single average values of orientations. Figure 3.7: Probability Distribution of the Direction Cosine. The probability distribution of the direction cosine from the surface normal of O-H vectors pointing into the vapor. 44

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Figure 3.8(a) displays the real a nd imaginary parts for the SSP spectrum calculated via Equations 3.6 and 3.7. Exam ining the real and imaginary parts of the spectrum can offer insights unavailable from th e modulus alone. The real and imaginary parts could be measured experimentally via a heterodyne detection scheme, or by taking advantage of interference effects between bulk and interfacial contributions to the spectrum [17]. To see the advantages of se parately examining the real and imaginary contributions, it is useful to write the re sonant SFG signal of a single harmonic mode, Q (with linear dipole and polarizabil ity), in frequency space as [4] 22 )2()( )/)(/()( IR IR jk i RQ Q, (3.11) 22 )2()( )/)(/()( IR jk i IQ Q. (3.12) In Equations 3.11 and 3.12, is a mathematical convergen ce parameter that physically can be interpreted as a homogeneous line widt h. The signal magnitude is seen to be proportional to the product of dipole and polarizability deri vatives. Equations 3.11 and 3.12 imply a single type of mode will lead to an imaginary contribution that is a symmetric well-defined peak (Lorentzian in ch aracter), while the real part will change sign, dipping below zero, at the maximum of the imaginary portion. If more than one species is contributing to the signal in a given region, a more complex lineshape will result from the overlapping signals. Examini ng the real and imaginary contributions in Figure 3.8(a), it is clear that several of the resonances are essentially single mode in 45

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Figure 3.8: Real and Imaginary Components of the Spectra. Real (s olid green line) and imaginary (dashed blue line) components of the (a) SFG SSP TCF spectra for the water/vapor interface and for (b) bulk water ca lculated as the Four ier-Laplace transform. character: the free O-H (3700 cm1), the small bending contribution at the surface (1800 cm1), the wagging mode (875 cm1), and translational modes (95 cm 1 and 220 cm1). There is some overlap in the translational m odes, and it is instru ctive that the higher frequency (220 cm1) mode, that is pronounced only in the SPS modulus spectrum, also shows up in the SSP real and imaginary spectra. Figure 3.9 highlights the O-H stretching regionfrom approximately 3000 cm1 to 3600 cm 1. Careful examination of the spectrum reveals three separate modes in this region centered at 3195 cm1, 3325 cm1, and 3400 cm1. Remarkably, this agrees very well with previous experimental work that deconvoluted the spectru m in this region. That analysis revealed three modes pres ent in the same region centered at 3200 cm1, 46

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3325 cm1, and 3454 cm1nearly the same frequencies [ 11,59]. This is strong evidence for distinct populations of water molecules in this donor O-H region of the spectrum. Figure 3.9: Real and Imaginary Components in the O-H Stretching Region. Real (solid green line) and imaginary (dashed blue line) components of the SFG SSP TCF spectra for the water/vapor interface for the O-H stretc hing region. The arrows highlight three separate modes centered at 3195 cm 1, 3325 cm1, and 3400 cm1. Further work is needed to identify the nature of these distinct O-H st retching species. It should be noted, that while the real and im aginary parts of the TCF derived SFG spectra do clearly indicate the presence of distinct subpopulat ions of oscillators, it is difficult to unambiguously identify the species responsible fo r the signals. This complication occurs because the modes are identifie d using INM methods, and the INM signal is broad in this region. Therefore, there is not an absolute correspondence between an INM frequency and the associated TCF spectrum (in this conge sted spectral region). This difficulty does 47

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not arise, however, in investigating spectra regions dominated by a single resonance like the free O-H or wagging mode. Further, the theoretical and experimental spectra have a somewhat different shape in this region, and this manifests itself in the relative intensities of the different contributions (considering the extant water/va por SFG spectra that have similar features but not identical shapes in th is region) [20,26,68,85,104]. The differences are most likely due to the spectroscopic intensities of these species via our spectroscopic model rather than different populations of these species at the interface within the MD model. However, further investigation is required to definitively demonstrate this. It should also be noted, as pointed out in an earlier investigation [6], orientational information can also be deduced from the relative signs of the imaginary mode lineshapes given knowledge of the signs of the prefactors in Equations. 3.11 and 3.12 (the dipole and polarizability derivatives). To further show the utility of the real and imaginary modal analysis, Figure 3.8(b) displays the real and imaginary parts of the bulk water O-H stretching region calculated as the FourierLaplace transform. While a linear IR experiment does not measure this observable, the transform can still be applied as an analysis tool. Figure 3.8(b) is strong evidence that there are two distinct species in the bulk, and the higher frequency moiety arises from the bulk free O-H, nonhydrogen bonded molecules [101,102,105]. 3.4 Conclusion The combined use of improved TCF and INM approximations to SFG spectroscopy represent a powerful complement ary approach. Achieving agreement with 48

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experimental measurements engenders confidence in the MD and spectroscopic models used to produce the theoretical spectrum. Many MD simulations of the water/vapor interface have been performed, but traditi onal analysis techniques do not easily uncover important interfacial subpopulations such as the wagging (hindered rotational) and hindered translational motions. Thus, SFG spectroscopy may be capable of giving a complete picture of the interfaceincludi ng structure and dynamics. Realizing this promise depends critically on the spectra being re liably interpreted, and the methods employed in this study are designed to unamb iguously characterize the nature of SFG spectra including inferring subpopulations of molecules from complex lineshapes. The plan is to investigate more complex and interesting interfaces using our improved combined INM/TCF approach. 49

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Chapter 4 SFG Spectroscopy of the Carbon Tetrachloride/Water Interface Theoretical approximations to the sum frequency generation (SFG) vibrational spectrum of O-H stretching at the carbon tetrachloride/wate r interface are constructed using quantum corrected time correlation f unction (TCF) and instantaneous normal mode (INM) methods. Detailed comparisons of th e theoretical signals are made with those obtained experimentally, and show good agre ement for the spectral peaks in the O-H stretching region of water. An intermolecular mode at 848 cm-1 is also identifiable, similar to the one seen for the water/vapor interface. Using INM methods, the resonance is seen to be due to a wagging mode that wa s previously identified [3] as localized on a single water molecule with both hydrogens displaced normal to the interface. Additionally, examination of the real and imag inary parts of the theoretical SFG spectra reveal the spectroscopic species attributed to the resonances in the O-H region, and are consistent with experimental data. 4.1 Introduction The interaction of water with hydrophobic surfaces is ubiquitous in chemistry and the environment, and plays an important ro le in many biological processes due to the unique molecular structure of water, which enables it to form an extended hydrogen 50

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bonding network [106]. Recent years have seen a large increase in the theoretical and experimental interest of the investigation into th e structural and dynamical properties of water at hydrophobic interfaces [insert refere nces]. The advent of interface specific nonlinear optical spectroscopies, such as SFG, have facilitated the und erstanding of such aqueous systems at the molecular level. The complementary nature of theory and experiment permits a more detailed understanding of the interface under study. In this chapter, classical molecular dynamics (MD) methods are applied to model the dynamics of the carbon tetrachloride/water interface. The configurations generated are then employed by the combined use of qua ntum corrected TCF and INM theories of vibrational spectroscop y in order to describe the SFG spectrum of the interface and determine the molecular origin of the SFG signal. The spectra obtained from this combined approach have been successfully compared with experimental spectra and demonstrates the effectiveness of these methods in understanding condensed phase spectroscopy of water, other liquids and interfaces [1-6,69-73]. 4.2 Models and Methods Using the methods described in the prev ious chapter, MD simulations were conducted using a code which uses reversible integration and extended system techniques [94]. Here, separate microcanonical MD simulations we re performed on ambient H2O with a density of 1.0 g/cm3 and CCl4 with a density of 1.5 g/cm3; both at an average temperature of 298 K. To create an inte rface, a cubic simulation box of equilibrated liquid water was placed between two equally sized simulation boxes of equilibrated liquid CCl4 along the z axis. The system was allowed to equilibrate, creating two carbon 51

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tetrachloride/water interfaces that were sufficiently far apart as to avoid strong interactions. Ewald summation was included in three dimensions, a nd the density profile of the system was monitored to verify equilibration [11]. In all cases, the results were tested and found to be system-size independent. Most results we re generated from 64 water and 52 CCl4 molecule simulations, and larger system sizes up to 128 water and 104 CCl4 molecules were tried, and did not alter the results. The MD simulations performed for this study were conducted using a carbon tetrachloride model that in cluded bond and bending potentia ls described by the Amber force field [107]. The partial charges on th e carbon tetrachloride atoms were fit to an ab initio calculated electrostatic potential surface using th e Connolly method in the GAMESS package [58]. As in previous work, the flexible simple point charge (SPC/F) water model that includes a harmonic bending pot ential, linear cross terms, and Morse OH stretching potentials was used [3]. For the purpose of modelling the spectroscopy, the partial point charges that were placed on th e water atoms were chosen to reproduce the condensed phase dipole moment [1-3,5]. As was seen with the water/vapor interface in the previous chapter, the interface is constructed using the standard MD geometry with CCl4 above and below the water [3,4,8]. This produces two interfaces, how ever, with average net dipoles in opposite directions and molecules at one interface that can diffuse to the other interface over time, perturbing the SFG signal [1-4]. To solve this problem, a weak (laterally isotropic) restraining potential was added to effectively confine the molecules to the half of the simulation box that they started in (normal to the interface) over the length of the 52

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simulation. The external potential was chosen such that it did not significantly perturb the relevant dynamics, as has been shown previously [3,4]. While the MD simulations were performe d without explicit polarization forces, polarizability is included when the SFG TCF or INM spectra are calculated. The induced dipoles and polarizability tensor of each configuration is de termined using a point atomic polarizability approximation (PAPA) model that includes many-body polarization effects explicitly, and accounts for polarizability deriva tives with point polar izabilities that are bond-length dependent [3,96-98]. The perman ent dipoles were calculated based on the same ab initio data [1,5] as described in the previous chapter. 4.3 Discussion Figure 4.1 displays the TCF derived theoretical description of the SFG spectrum for the entire water vibrational spectrum of the carbon tetrachloride/water interface in the SSP polarization condition. The theoretical spectrum has been adjusted in relative intensity to account for the Fresnel factors that modify the experimental intensities [3,15]. As was seen for the water/vapor interface [2,3], the inset of Figure 4.1 highlights an intense resonance in the intermolecular region identified here at 848 cm-1. Using INM methods [2-4], the resonance is shown to be due to the wagging mode, which is localized on a single water molecule, almost parallel to the interface, as described in chapter 3. Figure 4.2 presents the theoretical TCF SFG spectrum in the O-H stretching region for the SSP geometry. The free O-H peak is easily id entifiable at 3655 cm-1, while the rest of the stretching region has a more complicated shape. The inset of figure 4.2 53

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Figure 4.1: SFG TCF Spectrum for the CCl4/Water Interface. SFG SSP spectrum using the TCF method for the CCl4/water interface for the entire water vibrational spectrum. The inset shows just the intermolecular region. displays experimental data for the O-H st retching region using the same polarization condition [36]. The fr ee O-H stretching lineshape is captured by the theory, and the peak frequency is in good agreement with the experi mental data, while the rest of the spectrum in this region has similar features. Figure 4.3 displays the real and imagin ary parts of the SSP TCF spectrum in the O-H stretching region calculated using the esta blished methods [3,4]. Upon examination, three separate modes can be identifi ed in this region centered at 3210 cm-1, 3350 cm-1, and 3450 cm-1, in addition to the free O-H mode at 3655 cm-1. This is in good agreement with previous experimental work that dec onvoluted the spectrum in this region. That analysis revealed modes present in the same spectral region centered at 3250 cm-1, 3444 cm-1, and the free O-H peak at 3669 cm-1 [36,108], with only th e peak seen at 3350 cm-1 54

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in the theoretical spectrum uncorrelated. It should also be noted that the theoretical and experimental spectra have a somewhat different shape in this region (3200-3600 cm-1), and this manifests itself in the relative inte nsities of the different contributions. The Figure 4.2: SFG TCF Spectrum fo r the O-H Stretching Region of CCl4/Water. SFG TCF spectrum for the CCl4/water interface in the O-H stretching region for the SSP polarization condition calculated from theory (s olid red line). The inset is experimental data [36] for the same polarization geometry. experimental spectrum [36] is more pronounced on the blue side of the broad donor O-H region (to the red of the free O-H peak) compared to the theoretical result, and the subpopulations identified on that side of the lin eshape are relatively larger as well. The differences are most likely due to the sp ectroscopic intensities of these species via our spectroscopic model rather than different popul ations of these spec ies at the interface 55

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within the MD model. Howeve r, supplementary investigation is required to definitively demonstrate this. Figure 4.3: Real and Imaginary Compone nts in the O-H Stretching Region of the CCl4/Water Interface. Real (solid green line) and imaginary (dashed blue line) components of the SFG SSP TCF spectra for the CCl4/water interface for the O-H stretching region. The ar rows highlight three separa te modes centered at 3210 cm1, 3350 cm1, and 3450 cm1. Comparison with previous theoretical spectra obtained for the water/vapor interface [2,3] shows that some species are co mmon to both interfaces. This should not be considered surprising, since carbon tetrachloride is a hydrophobic medium and therefore creates a small buffer region, thus reducing the interference to the water structure at the surface. This can be seen in Figure 4.4. However, the intensity of the free O-H peak is reduced as compared to the water/vapor spectrum due to the interactions between the free O-H oscillat ors at the interface and car bon tetrachloride. Further 56

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examination of the two systems reveals that th e frequency of the O-H vibration is slightly lower for the CCl4/water interface (3655 cm-1) than for the water/va por interface. This red shift is consistent with experimental data, and has been found to be due to an attractive interaction between the dangling O-H bond and the surrounding carbon tetrachloride molecules, which hinders its motion [36,108,109]. Figure 4.4: CCl4/Water Interface Snapshot. A snaps hot of a carbon tetrachloride/water interface containing 64 water molecules and 26 CCl4 molecules on each side, featuring INMs from different regions of the spectra. The water molecule shown in blue is representative of a fr ee O-H mode at 3656 cm1. The water molecule shown in yellow is representative of a wagging motion at 848 cm1. Figure 4.4 highlights the representative vibrational modes from the O-H stretching and wagging regions that give rise to their respective spectral signatures. A typical free O-H mode, shown in blue, produces the high fr equency feature at 3655 cm-1. It is clear that the oxygen atom is anchored in the interface, while the O-H is oscillating freely above the interface. Th e intermolecular vibrational m ode identified as the wagging 57

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feature at 848 cm-1 is highlighted nearby in yellow. Here, the oxygen atom is anchored in the interface, and the two hydrogens are vibrating away from the CCl4 interface. The carbon tetrachloride molecules are shown in gr een. These results demonstrate how the INM method reveals physical characteristics of the interfacial modes, and how different molecular motions contribute to the spectrum. 4.4 Conclusions The vibrational SFG spectrum of the car bon tetrachloride/water interface has been calculated based on molecular dynamics si mulations in order to gain a better understanding of the interfacial structure of this system. A combined approach which makes use of improved TCF and INM approxim ations to SFG spect roscopy was used in the analysis. The spectrum obtained has been examined and compared to both experiment [36,108,109] and previous studi es of the water/vapor interface [2,3]. Achieving agreement with experimental measurements engenders confidence in the MD and spectroscopic models used to produce the theoretical spectrum, and suggests that our understanding of the spectroscopy of interfaces is improving. Many MD simulations of the carbon tetrachloride/water interface have been performed, but traditional analysis techniques do not easily uncover important interfacial subpopulations such as the wagging (hindered rotational) motions. Further analysis reveals that the frequency of the free O-H and wagging vibrations are slightly red shifted for the CCl4/water interface (when compared to water/vapor) due to interactions at the interface between CCl4 and water. Thus, SFG spectroscopy may be capab le of giving a complete picture of the interfaceincluding structure and dynamics. The methods employed in this study are 58

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designed to unambiguously characterize the nature of SFG spectra including inferring subpopulations of molecules from complex lines hapes. Additional investigations using the combined INM/TCF approach on other co mplex and interesting systems will provide more information about the intera ctions at hydrop hobic interfaces. 59

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Chapter 5 Third Order SFVS of the Silica/Water Interface Sum frequency vibrational spectroscopy (SFVS), a second order optical process, is interface specific in the dipole approxi mation [4,23]. At charged interfaces, the experimentally detected signal is a combination of enhanced second order and static field induced third order contributions due to the existence of a st atic field. Evidence of the importance/relative magnitude of this third or der contribution is seen in the literature [110,111,112], but no previous molecularly detailed appro ach existed to separately calculate the second and third order contributions. Recent work presented a novel molecular dynamics (MD) based theory that provides a direct means to calculate the third order contributions to SFVS spectra at char ged interfaces [113], and a hyperpolarizability model for water was developed [114]. Here, these methods are applied to an idealized silica/water interface, an d the results are compared to ex perimental data for water at a fused quartz surface. 5.1 Introduction Aqueous interfaces are abundant in the e nvironment, and vital to many chemical, biological, and atmospheric processes. T ypically, these interfaces are not neat, and contain charged species or charged solid surf aces, such as silica. Naturally occurring 60

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silica/water interfaces are ubiquitous, and many important chemical processes occur in soil and atmospheric dust, where silicates ar e widespread [115-119]. A charged interface is created between water and silica by the presence of silanol groups (SiOH), which terminate silica and tend to i onize in water, especially as pH is increased [110,111,120]. A relatively large static fiel d is also produced by the undi ssociated silica surface due to large charge separation between the s ilicon, oxygen, and hydrogen atoms [120,121]. In a typical SFVS experiment, a signal is generated by comb ining visible and infrared laser pulses focused upon the interface to be studied. The resulting signal is a second order response which requires anisotropic media according to the dipole approximation [4,5,15,23]. Interfaces inherently disrupt the symmetry due to the lack of an inversion center which gives rise to th e interface specificity needed. When the infrared laser frequency corresponds to a vibra tion at the interface, a resonant lineshape is obtained with a characteristic shape that reflects both the structural and dynamical environment at the interface [4,15,45]. If a static field is present, the third order susceptibility is probed in addition to the seco nd order susceptibility. Due to the static nature of the third field, the second and thir d order signals are generated in the direction given by the sum frequency wave vector and are inseparable experimentally [110,111,114]. Recently, a computational molecularly deta iled method to model both the second and third order contributions to the SFVS signal of charged interfaces was developed [113]. Previous theoretical studies [1,2 ,3,5,72,73,74,122], including the ones presented in chapters 3 and 4, used a semi-classical t echnique to calculate quantum corrected TCFs that describe the vibrational spectra of comp lex liquids and interfaces. Briefly, classical 61

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MD was performed to generate trajectories for an aqueous system. The permanent dipoles, polarizabilities, and their deriva tives for each species present in the MD simulation are parametrized as a func tion of molecular geometry based on ab initio data. In order to construct the corresponding TC Fs, the induced dipoles and polarizability tensor arising from the interatomic interactio ns at each configuration are then calculated using a point atomic polarizability approxima tion (PAPA) model. This approach has been demonstrated to be an effective method for understanding the condensed phase spectroscopy of water and othe r liquid interfaces [1,2,3,72, 73,74], and has been extended to calculate the third order contribution to SFVS spectra at charged interfaces [113,114]. Here, this extended spectroscopic model is app lied to MD generated configurations of an idealized silica/water interface, in order to determine if the new model is suitable for evaluating third order optical effects at more complex charged interfaces. 5.2 Models and Methods The system modeled during the course of this work includes a charged surface at an interface with water. T ypical three-wave mixing experm inents, such as SFVS, probe both the second order, (2), and third order, (3), susceptibilities in the presence of a static field created by a charged species [4,17, 110,111,123-125]. The third order polarization term can be shown to depend on an additional contribution fr om the electrostatic field, Estatic, to the nonlinear polarization induced at the interface by the visible, Evis, and infrared, EIR, fields, and is given by [24,113,110,111] staticIRvis IRvis SFGEEEEE PPP)3( )2()3()2(: (5.1) 62

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Here, P(2) and P(3) denote a second and third order polar ization, respectively. Equation 5.1 therefore implies that the measured pol arization signal contains the normal second order SFG signal represented by the first term on the right hand side, in addition to the third order contribution shown in the second term, which contains the presence of three fields (two incident and one static). This contribution is not negligible [17,24,110,111,129], and since the electrostatic field disrupts the symmetry of the system through the alignment of interfacial water mol ecules, it serves to extend the anisotropic interfacial region into normally centrosymme tric regions of the bulk [17,26,35,130-132]. This allows more molecules to contribute to the observed nonlinea r polarization [24,113]. Due to the wave vector of the static field being zero, the observed polarization signal, PSFG, in Equation 5.1 represents a combin ed measurement of second and third order processes. The wave vector of a fi eld is proportional to its frequency, and the experimentally detected polari zation signal is determined by the sum of the wave vectors, ks, from the excitation fields [4,25,87]. Ther efore, since the two applied fields have nonzero wave vectors, k1 and k2, while the static field has a wave vector of zero, the direction that the generated signal will propagate is represented by ks = k1 + k2 and ks = k1 + k2 + 0 for the P(2) and P(3) signals, respectively. To investigate the third order nonlinear polarization contribution in molecular detail computationally, it is necessary to st art with perturbation theory [126]. It has previously been shown that the 48 terms used to describe the third order susceptibility can be simplified in terms of the real part of two different TCFs [113]. The dominant 63

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contribution is given by cross-correlating the systems dipole and hype rpolarizability, and is defined as dtt eikzj ti IR s kjizIR})0()(Im{ )(0 2 Re),3( (5.2) In Equation 5.2, the imaginary part of th e equation is represen ted by the designation Im while denotes the system hyperpolarizability, and z is the component perpendicular to the interface that is enhanced by the static field. The time correlation function in this equation is proportional to kzj i, by )0()()(ikzjttG (5.3) It can then be shown that analysis using detailed balance reveals that the Fourier transform of the real, GR(t) and imaginary, GI(t), parts of the TCF are analytically related by a factor of tanh as )()2/tanh()( R IG G (5.4) where GR() and GI() are both real. As described in previous work [70,72], the classical limit can then be found and the corr elation function can be quantum corrected. Application of the TCF described (to th e modeling of aqueous charged interfaces) requires the modification of the established spectroscopic mode l [3,4], that was used in chapters 3 and 4, to include the calculation of the molecular and system 64

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hyperpolarizabilities. This will enable the cont ribution of the third order effects resulting from the presence of a static field to be determined. To construct the spectroscopic model, point polarizabilities are assigned to each atom, such that the interactions reproduce the equilibrium gas phase hyperpol arizability tensor [127,128]. Since the dependence on molecular geometry is included explicitly in the spectroscopic model, the hyperpolarizabilities and their derivati ves were determined from fits to ab initio electronic structure calculations [113,114]. Methods using appropriate basis sets have shown reliability in determining the static hyperpolarizability of gas phase and liquid water [137-141]. Additionally, this method us es an Applequist/Thole-like model [1-4,7374,100,133-136], to calaculate the effective hyperpolarizabili ty, including intrinsic and interaction effects, as sums over the products of the condensed phase polarizability and matrices related to the dipole interation tensor This process is similar to the many body polarization method that was used to calculate the polarizability in chapters 3 and 4. Using the Applequist/Sundberg formalism, the total effectiv e hyperpolarizability, is given by [128] eff ijk n n eff nk n eff nj n eff ni n eff ijk (14) where n denotes the intrinsic, internal molecula r geometry dependent hyperpolar izability associated with atom n The total effective polarizability between atom i and n is represented by and n is the intrinsic, internal molecular geometry dependent polarizability associated with atom n eff ni 65

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As in the previous chapters, MD simula tions were performed using a code that uses reversible integration and extended system techniques [94]. Microcanonical MD simulations were performed on ambient H2O with a density of 1.0 g/cm3 and an average temperature of 298 K. To create an inte rface, two equally sized cubic simulation boxes of equilibrated liquid water were placed on opposite sides of an idealized flat surface along the z axis. As can be seen in figure 5.1, this idealized surface was constructed as a rigid slab of double layer charges, where one side of the slab contains a negatively charged layer followed by a positively char ged layer, while th e opposite side is assembled in reverse. Although this system is an idealized approximation, it is similar to glasses, such as fused quartz, and is an excellent precursor to modeling more complex water/silica interfaces. The system was allo wed to equilibrate, cr eating two water/solid interfaces that were sufficiently far apart as to avoid strong interactions. Most results Figure 5.1: Silica/Water Inte rface Snapshot. A snapshot of the idealized silica/water interface containing 242 water molecules and on each side. The silica slab is comprised of 378 oxygen atoms, shown in red, and an equal number of silicon atoms, shown in yellow. 66

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were generated from 484 water molecule si mulations. Ewald summation was included in three dimensions, and the density profile of the system was monitored to verify equilibration [11]. The MD simulations performed for this study were conducted us ing a rigid silica slab model that included intermolecular interaction potentials described by the Universal Force Field [142]. The partial charges on the silica atoms were selected to maintain charge neutrality for the system. As in prev ious work, the flexible simple point charge (SPC/F) water model that includes a harmoni c bending potential, linear cross terms, and Morse O-H stretching potenti als was used [3]. For the purpose of modelling the spectroscopy, the partial point charges that were placed on the water atoms were chosen to reproduce the condensed phase dipole moment [1-3,5,114]. 5.3 Discussion Figure 5.2 shows the theoretical descrip tion of the SFVS spectrum derived using the third order TCF method for the entire wa ter vibrational spectrum of the idealized silica/water interface in the SSP polarization condition. When compared with the spectra obtained for the water/vapor and CCl4/water interfaces shown in chapters 3 and 4, it can be seen that the spectrum is now dominated by the features in the O-H stretching region. The resonance seen around 1800 cm-1 is relatively more intense (as compared to the previous interfacial spectra), and is due to bending motion of bulk water molecules near the interface [3]. This increase in intensity indicates that the interface has been extended further into the bulk ( i.e. disrupting the symmetry) by the presence of the charged 67

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surface, thus allowing a greater contribution to the SFVS spectra from this typically bulklike behavior. Figure 5.2: Third Order SFVS Spectrum for the Silica/Water Interface. SFVS spectrum using the third order TCF me thod in the SSP polarization condition for the idealized silica/water interface fo r the entire water vibrational spectrum. Figure 5.3 presents the theoretical thir d order TCF SFVS spectrum in the O-H stretching region for the SSP pol arization condition. Here, tw o peaks in the area between 3100 3500 cm-1 are clearly identifiable. The first peak at approximately 3200 cm-1 can be attributed to ice-like ordering of the wa ter molecules near the interface, while the second peak that occurs around 3400 cm-1 is more liquid-like, and is similar to the peak seen in the donor O-H region of the wa ter/vapor spectrum [4,17,23,112,143-145]. 68

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Figure 5.3: Third Order SFVS Spectrum for the O-H Stretching Region of the Silica/Water Interface. The third order TCF SFVS spectrum for the idealized silica/water interface in the O-H stretching region for the SSP polarization condition calculated from theory (solid red line). The inset is experime ntal data [143] for fu sed quartz/water using the same polarization geometry. The free O-H peak, which normally occurs around 3600 cm-1, is clearly suppressed, which indicates that most of the water molecules are hydrogen bonding at this surface [17,143-145]. The inset of Figure 5.3 displays experimental data fo r the O-H stretching region of a fused quartz/water interface taken in the same polarization geometry [112,143]. The relative intensities agree ne arly quantitatively between theory and experiment, and the line shape is captured remark able well by the theory. This indicates that the third order spectros copic model developed [113,114] is capable of reproducing the essential features of the spectrum. 69

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Figure 5.4: Second Order SFVS Spectrum for the O-H Stretching Region of the Silica/Water Interface. SFVS spectrum for the idealized silica/water interface in the O-H stretching region for the SSP polarization condition using th e second order TCF model. For comparison with the third order model, Figure 5.4 shows the spectrum for the same system and polarization geometry in th e O-H stretching region using the established second order TCF expression [1-4], presented in chapter 3. Using this method, the peaks appear less well defined, and the signal intens ity is reduced by an order of magnitude. While more refinement of the model is need ed, this comparison suggests that the third order contribution does play an important role in determining the spectra of a charged interface. In Figure 5.5, the theoretical third order TCF SFVS spectrum in the O-H stretching region for the SSP polarization co ndition is shown for an idealised silica 70

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interface where the negative surface charge has been in creased. Introducing an artificially high negative surface charge to th is interface approximates a fully ionized Figure 5.5: Third Order SFVS Spectrum for the Silica/Water Interface with Increased Surface Charge. The third order TCF SFVS spectrum for the id ealized silica/water interface with high surface ch arge in the O-H stretching region for the SSP polarization condition calculated from theory (solid red line). The inset is experimental data [143] for fused quartz/water using the same polarizat ion geometry at a pH level of 12.3. silica surface, producing a strong electric field. Both experiment and theory show that this results in a more highly ordered interfacial st ructure, as evidenced by the dominance of the ice-like peak at 3200 cm-1 [17,112,143,144]. This indicates that the fiel d can orient several molecular layers of the water inte rface with one or bot h hydrogens pointing toward the silica interface to form a hydroge n-bond with the deprot onated surface. The 71

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appearance of the should er feature near 3400 cm-1 results from the more liquid-like structure as the effects from the electric field are not as strong going into the bulk. 5.4 Conclusions Theoretical approximations to the in terface specific SFVS spectrum of O-H stretching at the water/sili ca interface have b een calculated based on molecular dynamics simulations of water at an idealized silica slab surface. Spectra are constructed using newly developed static field enhanced TCF methods [113,114]. This approach leads to a signal in the SSP polarization geometry th at is comparable with experimental measurements [112,143]. SFVS is an inhere ntly challenging second order optical technique and studies on water interfaces concentrate on the line shape associated with the intense O-H stretching resonance th at is perturbed by the inhomogeneous environment. The presence of a static fiel d associated with the charged species can produce an apparent ice-like or dering of water molecules in the OH stretching region at the interface, which corresponds to published e xperimental data at interfaces with fused quartz [112,143,144] and -quartz [17,144]. The complexity of the broad structured SFVS signal can be attributed to O-H stretchi ng motions facing toward the bulk or silica surface environments that are characteristic of the interface. Achi eving agreement with experimental measurements engenders conf idence in the MD and spectroscopic models used to produce th e theoretical spectrum, and suggests th at theory can play a crucial role in interpreting SFVS spectra at more comp lex interfaces. Further investigation and refinement of this model is clearly required, but the initial results are promising, and build on previous success in understandi ng the complex structure and dynamics of 72

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aqueous interfaces. Additionally, future work may determine the level of importance of third order signals, and whether ordering of the interfacial wa ter structure contributes to an enhanced second order signal. 73

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Chapter 6 Computer Simulation of Molecular Evolution This chapter presents a molecular evol ution code affectionately named EVO, which has been developed to investigate evolutionary relatedness among various groups of organisms using computational phylogeneti cs methods. This program simulates the mutation of a nucleic acid or protein coding sequence along several branch lines, and creates two output files formatted for use in the software packages Molecular Evolutionary Genetics Analysis (MEGA) and Clustal. The s ource code of this simulation package is listed in Appendix B. Molecular phylogenetics methods rely on a defined substitution model that encodes a hypothesis about the relative rate s of mutation at vari ous sites along the sequences being studied [146,147] Variations in the time increment between mutations, the number of branches, the site insertion/de letion rate, and the site substitution rate can be controlled through parameters specified in th e input files. The main input file contains information about the sequence to be mutated, including sequence identity, number of mutations, branch time, and initial rates for the substitutions and insertion/deletions. Additional input files contain information about the variations in the rate at which the insertion/deletions and substitutions occur at each site in the sequence. Figure 6.1 shows 74

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a graphical user interface (GUI) that was developed to fac ilitate the use of the EVO executable by students. Figure 6.1: Interface for the EVO Simulatio n Package. The EVO simulation package was developed as a tool to investigate mol ecular phylogenetics. The GUI displays four fields, which allow the user to search for input files and a location to save the output files. The files used to define the variations in the insertion/deletion and substitution rates can be created using a complementary code that was developed named Site Rate. 75

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The rates of variation are ca lculated based on a gamma distribution algorithm first developed by Cheng and Feast [148]. As can be seen in Figure 6.2, a GUI has been created, which allows the user to easily defi ne certain parameters used to generate the distribution. Appendix C presents the source code for the Site Rate file generator package. Figure 6.2: Interface for the Site Rate Package. The Site Rate file generator package creates an input file for use in EVO that contains information about the variations in sequence site change. The site rate variation data created is then used in the EVO package to calculate the rate at which an individual site might e xperience either an insertion/deletion or a 76

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substitution, since this probability is not e qual for all sites along a sequence [146,149]. If a substitution is determined to occur, the nuc leotide designation is then changed to one of the other three possibilities [149] based on a random number generator that uses a shuffling algorithm presented by Press et al [150]. Insertions and deletions for a site are resolved using a similar approach, and in the case of an insertion, the nucleotide designation is determined by employing the same random number generator algorithm. Output from the simulation is intended for use in the MEGA and Clustal software packages, which conduct multiple sequence alignment and infer phylogenetic trees, with the purpose of testing evolutionary hypotheses [151,152]. Additionally, a software tool was developed to search for an chor points in the sequences ge nerated, in order to further investigate evolutionary relationships. A dditional input options can be specified to modify the sequences, if necessary, as can be seen in Figure 6.3. The source code of this package is available in Appendix D. Further development of the codebase includes the capability to make additional adjustments specific to protein coding sequen ces. Rates of mutation can be adjusted for the position of a given site w ithin a codon to allow for highe r mutation rates in the third nucleotide of a given codon without affecting the meaning of the codon in the genetic code [147]. Supplementary refinement of th e interface is also in corporated into the package, and implements more on-the-fly interaction with the user. 77

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Figure 6.3: Interface for the Anchor Search Package. The Anchor Search package uses output files from EVO to search for anchor points in the mutated sequences generated. 78

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Chapter 7 Conclusion Molecular theory and computer simulation have proven to be essential resources for exploring and understanding the chemical and physical properties of a variety of systems, including aqueous interfaces. Mo lecular dynamics, time co rrelation function, and instantaneous normal mode techniques de monstrate a compelling potential to provide a better comprehension of experi mental results at the molecular level. This can clearly be seen in the investigations presented here for the calculation of molecular volumes and vibrational spectra, and in numerous published articles. Simulations of molecular dynamics using the isobaric-isother mal (NPT) ensemble can effectively be used to calculate time-dependent molecular volumes in conjunction with photothermal experiments, as was demonstr ated in chapter 2 for the isomerization of azobenzene. Computational tools ( i.e. simulation snapshots) convey detailed microscopic information about the structural changes and dynamics accompanying molecular volume changes. This investigation is in excellent agreement with experimental results, and engenders confidence in the ability of these th eoretical methods to provide an atomistic level of understanding of the forces and struct ural factors that cont ribute to molecular volume changes. 79

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80 The information obtained from interfacespecific nonlinear spectroscopic methods combined with theoretical m odels provides valuable insight about the structure and dynamics at interfaces. The studies outlined in this work aim to improve the accuracy of the models in use today, with the goal of gaining a complete picture of aqueous interfaces. Reliable interpretation of vibrational spectra is critic ally important to this end. The methods described here are therefore capable of character izing the spectra explicitly, which includes the identification of molecu lar species from complex lineshapes. Aqueous interfaces are abundant in chem ical, biological, and environmental processes, and the investig ation of such fundamentally important systems using theoretical methods facilitates the interpretation of data obtained experimentally. A dynamic relationship between theory and experi ment is needed to further develop the analytical and predictive capability of theo retical studies. The in vestigation of more complex interfaces using the MD, TCF, and INM methods described in this work will assist in both the interpretation of the larg e and expanding quantity of experimental data, and the prediction of previously unexplored vibrational structures at interfaces.

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Appendices 91

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Appendix A: Time Domain Expressi on for the Sum Frequency Response Starting from frequency domai n perturbation theory for six terms are obtained, and are shown below [5,25,87]. Four of the terms contribu te to the resonant signal (contained in R1 and R2 below), and two are nonresonant (NR1 and NR2 below). While two of the resonant terms may appear initially to be nonresonant, those with denominators containing the expression (IR+ng+ing), they contribute to the resonant susceptibility, and lead to the co mplex conjugate correlation function C*(t). Ultimately, their inclusion is necessary to reproduce Equation 3.4. In the expressions below, IR and vis are the frequencies of the infrared and visible fields. While, SFG is the sum frequency of the infrared and visible fields, and ng is the frequency corresponding to the energy difference between energy levels n and g. In Equation A1, is the initial state thermal population, and the sum is over vibronic levels. Here, is a dipole matrix element between states and for dipole vector component : )()2(SFG)0( g mng g IRvisSFG pqrNRNRRR,, 2121 )0() )((),,( (A1) ) () () (1 mg ngIR r gn ng ng vis p nm q gn ng ng SFG q nm p gni i i R ) () () (2 ng ngIR r gn mg mg vis q mg p nm mg mg SFG p mg q nmi i i R ) )( (1 ng ng vismg mg SFG r nm p mg q gni i NR 92

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) )( (1 mg mg visng ng SFG r nm q mg p gni i NR The resonant contributions can be simplified by rewriting in terms of polarizabilities and dipoles, and the approximation 1/SFG = 1/vis. Given the definition of polarizability in Equation A2, the two resonant terms, R1 and R2, simplify to Equations A3 and A4, respectively: )0( ,)(g ng ng ng p ng q gn ng ng q ng p gn pqi i (A2) ) (1 mg mg IR r mg pq gmi R (A3) ) (2 ng ngIR pq ng r gni R (A4) Let s pqr Re denote only the sum of the resonant terms R1 and R2. Replacing the denominators in both of the resonant terms with the integral identities and and then taking the explicit limit that gamma goes to zero gives Equation A5. Equation A6 follows an exact rewrite of Equation A5, and expresses the susceptibility in terms of the cross correlation of the system dipole and polarizability: ) /(0 0 (ii edtit )0i) /(0 0 )(0ii edtiit 93

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)0( 0 0 Re g ng r gn pq ng ti ti gm r mg pq gm ti ti s pqrdt ee i dt ee iIR ng IR mg (A5) )()0( )0()(0 0 Ret edt i tedt ipqr i rpq i s pqrIR IR (A6) Expansion of the correlation functions in Equation A6 results in a significant simplification. Noting, r(0) pq( t ) = CR( t ) + iCI( t ) = (pq( t ) r(0))* gives Equation A7 belowwhich is given in Chapter 3 as Equation 3.4. 0 )( Re)( 2 )( tCedtI it IR s pqrIR (A7) 94

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Appendix B: EVO Simulation Package ################################ # !!! common Makefile !!! # ################################ CPP = c++ CPPSRCS = evo.cpp random.cpp gamma.cpp all: evotest evotest: evo.o random.o gamma.o $(CPP) -g -o evotest evo.o random.o evo2: evo.o random.o gamma.o $(CPP) -o evo2 evo.o random.o clean: @echo "'make clean': removing compiled executables and object code" -/bin/rm *.o evotest evo2 /* evo.h */ #include #include #include #include #include #include #include // #include // #using using namespace std; int evo(char* infile1, char* infile2, char* infile3, char* outfile1); float ran1( long *idum); float gammadev( int ia, long *idum); /* random.h * Random Number Generator ran1() from Numerical Recipes called from mutate() in evo. It returns a random number between 0 and 1. * idum is the random seed generated in evo */ float ran1( long *idum); 95

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/* random.cpp * Random Number Generator from Numerical Recipies. It returns a random number between 0 and 1. * usage: float ran1(long *idum) * idum is the random seed generated before the function call. */ #include "evo.h" #define IA 16807 #define IM 2147483647 #define AM (1.0/IM) #define IQ 127773 #define IR 2836 #define NTAB 32 #define NDIV (1+(IM-1)/NTAB) #define EPS 1.2e-7 #define RNMX (1.0-EPS) float ran1( long *idum) { int j; long k; static long iy=0; static long iv[NTAB]; float temp; if (*idum <= 0 || !iy) { if (-(*idum) < 1) *idum=1; else *idum = -(*idum); for (j=NTAB+7;j>=0;j--) { k=(*idum)/IQ; *idum=IA*(*idum-k*IQ)-IR*k; if (*idum < 0) *idum += IM; if (j < NTAB) iv[j] = *idum; } iy=iv[0]; } k=(*idum)/IQ; *idum=IA*(*idum-k*IQ)-IR*k; if (*idum < 0) *idum += IM; j=iy/NDIV; iy=iv[j]; iv[j] = *idum; if ((temp=AM*iy) > RNMX) return RNMX; else return temp; } 96

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#undef IA #undef IM #undef AM #undef IQ #undef IR #undef NTAB #undef NDIV #undef EPS #undef RNMX /* evo.cpp version 2.1.0 * Tony J. Green Space Research Group Department of Chemistry University of South Florida */ /* This program simulates the mutation of a protein or nucleic acid sequence. Output is in two files formated for use in the programs MEGA (.meg) and Clustal (.pir). */ //-------------------------header files ------------------------------#include #include #include #include #include #include #include "evo.h" using namespace std; // ----------------------------------------------------------------------// structure which stores the information about the sequences struct masterp { int localtime; // stores the local position char seq[4000]; // stores the sequence int branchtime ; // stores the branch time of each sequence int localpos; // stores the local position int alpharate; // stores the alpha rate for each sequence int indelrate; // stores the insertion/deletion rate for each sequence char seq_id[100]; int parentbranchtime; }sysinfo[100]; // declaration of the global variables int no_of_branches = 0; // stores the total number of branch count 97

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double *indelgam; // array that holds variation multiple for indel sites double *alphagam; // array that holds variation multiple for subst. sites int length; // the length of the main sequence name int btime[100]; int currptr[400]; // keeps track of the sequences which are active int num_active; // keeps track of the number of active sequences int globcount; // keeps track of the position of the currptr int cntr = 0; int alpharate; // stores the alpha rate of the the sequence int indelrate; // stores the indel rate of the sequence int flaginsert; // variable which stores whether insertion took place int seqlength; // variable which stores the length of the sequences int flag = 0; // flag variable ****** int totaltime = 20; static int count = 0; // static to count the sequence numbers int insertion[40]; int mask[1500]; char dum; char megafile[5000]; FILE *fp_pir; FILE *fp_std; FILE *fptest; // declaration of subroutines void initialize( long *idum); void simulate( long *idum); void mutate( long *idum); void branch( int x); void shift( char *a, int pos); int searchcurrptr( int x); void updatecurrptr( int x); void convert( char *, char *); void reset(); /* Resets the global variables to their initial conditions at the begining of each execution. Prevents value retention while the gui keeps the variables in limbo. called from evo() below */ void reset() { no_of_branches = 0; cntr = 0; flag = 0; totaltime = 20; count = 0; 98

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} /* Initializing function which creates the first (master) sequence called from evo() below */ void initialize( long *idum) { int i, j; // local variables char masterseq[5000]; // to store the sequence locally num_active = 1; // point to the number of active sequences currptr[0] = 0; // points to the number of active sequences printf("\n\nEntering the initialize procedure ...\n"); for(i=0; i<1200; i++){ j = 1+( int) (4.0*ran1(idum)); // get random number from 1 to 4; // add a number for the offset switch (j) { case 1: masterseq[i] = 'A'; break; case 2: masterseq[i] = 'C'; break; case 3: masterseq[i] = 'T'; break; case 4: masterseq[i] = 'G'; break; default : masterseq[i] = '#'; } } masterseq[i] = '\0'; strcpy(sysinfo[0].seq,masterseq); for(i = 0 ; i < 100 ; i++){ sysinfo[i].localpos = 0; sysinfo[i].localtime = 0; sysinfo[i].parentbranchtime = 0; } } /* Start Of the main subroutine */ int evo(char* infile1, char* infile2, char* infile3, char* outfile1) { int i,j; long idum; FILE *fp_in1; // for opening the sequence data file FILE *fp_indel; // for opening the indel rate variance file 99

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FILE *fp_alpha; // for opening the subst. rate variance file char seq[100]; seqlength = 1200; // stores the sequence length reset(); // called to reset some globals srand( time(NULL) ); // generate random seed here once for entire program idum = -( time(NULL) ); // initial seed to use for ran1() strcpy(megafile,outfile1); strcat(outfile1,".txt"); fp_in1 = fopen(infile1,"r"); // opens input parameter file for reading fp_std = fopen(outfile1,"w"); // opens stdout txt file for writing fgets(seq,100,fp_in1); length = strlen(seq); fprintf(fp_std,"The value of length is %d\n",length); // print to stardard text file fclose(fp_in1); fp_in1 = fopen(infile1,"r"); for(i = 0 ; i < (2*length 3) ; i++) { fscanf(fp_in1,"%s",sysinfo[i].seq_id); fprintf(fp_std,"sequence id = %s\n",sysinfo[i].seq_id); fscanf(fp_in1,"%d",&sysinfo[i].branchtime); fprintf(fp_std,"branch time = %d\n",sysinfo[i].branchtime); btime[i] = sysinfo[i].branchtime; fscanf(fp_in1,"%d\n",&sysinfo[i].indelrate); fprintf(fp_std,"Indel rate = %d\n",sysinfo[i].indelrate); fscanf(fp_in1,"%d",&sysinfo[i].alpharate); fprintf(fp_std,"Alpha rate = %d\n\n",sysinfo[i].alpharate); } fclose(fp_in1); fp_indel = fopen(infile2,"r"); fp_alpha = fopen(infile3,"r"); indelgam = ( double *)calloc(seqlength+1, sizeof ( double )); alphagam = ( double *)calloc(seqlength+1, sizeof ( double )); for (j=0; j
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fclose(fp_alpha); length = length 1; // just a test, otherwise it is equal to the length of the main sequence fprintf(fp_std,"rand max = %d\n",RAND_MAX); fflush(stdin); initialize(&idum); simulate(&idum); fclose(fp_std); free(indelgam); free(alphagam); return 0; } // end of main /* function which starts simulation and opens output files for writing called from main() above */ void simulate( long *idum) { FILE *fp_meg; char pirfile[5000]; char meg[10],pir[10]; flag = 0; strcpy(meg,".meg"); strcpy(pir,".pir"); strcpy(pirfile,megafile); strcat(megafile,meg); fp_meg = fopen(megafile,"w"); // opens .meg file strcat(pirfile,pir); fp_pir = fopen(pirfile,"w"); // opens .pir file fprintf(fp_meg,"#MEGA\n"); // printed to .meg fprintf(fp_meg,"Title:\n"); // printed to .meg int j=0,k; int branch_seq[4000], branch_ptr; int num_active2; int removeptr; int remove_seq[4000]; fprintf(fp_std,"\n branchtime[10] = %d",sysinfo[10].branchtime); // stdout print fptest = fopen("RateInfo.txt","w"); while(num_active > 0) { branch_ptr = 0; 101

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removeptr = 0; mutate(idum); // call function mutate() fprintf(fp_std,"\n"); // screen print for(j=0; j < num_active; j++) { sysinfo[currptr[j]].branchtime = btime[currptr[j]]; fprintf(fp_std,"\n %d: branchtime = %d, localtime = %d.",currptr[j],sysinfo[currptr[j]].branchtime, sysinfo[currptr[j]].localtime); // screen print if(sysinfo[currptr[j]].branchtime == sysinfo[currptr[j]].localtime) { fprintf(fp_std,"\n branch sequence[%d]: %s",currptr[j],sysinfo[currptr[j]].seq_id); // screen print branch_seq[branch_ptr] = currptr[j]; branch_ptr++; } } if(branch_ptr == 0){ for(j=0; j < num_active;j++) sysinfo[currptr[j]].localtime++; } num_active2 = num_active; for(j=0; j < branch_ptr; j++) { if(strlen(sysinfo[branch_seq[j]].seq_id)== 1) { fprintf(fp_meg,"#"); // print to .meg fputs(sysinfo[branch_seq[j]].seq_id,fp_meg); // put seq. number after # fprintf(fp_meg," "); // print to .meg fputs(sysinfo[branch_seq[j]].seq,fp_meg); // put full seq. after 15 spaces fprintf(fp_meg,"\n"); // print to .meg convert(sysinfo[branch_seq[j]].seq,sysinfo[branch_seq[j]].seq_id) ; // call function convert() } num_active--; if(flag == 0){ branch(branch_seq[j]); // call function branch() fprintf(fp_std,"\n Branching sequence %s ",sysinfo[branch_seq[j]].seq_id); } 102

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else { remove_seq[removeptr] = branch_seq[j]; removeptr++; } if(num_active == 0) break; } for(j = 0; j < removeptr; j++) { fprintf(fp_std,"\n remove %s",sysinfo[remove_seq[j]].seq_id); // screen print k = num_active2; while((k >= 0) && (currptr[k] != remove_seq[j])) k--; while(k <= num_active2) { currptr[k] = currptr[k+1]; k++; } } fflush(stdin); } fclose(fp_meg); fclose(fp_pir); fclose(fptest); } /* function which contains the insertion deletion and subsitution logic called from simulate() above */ void mutate( long *idum) { int a,b,dummy; int i,j,k,l; int d,rand1; int insert = 1; int count = 0; char brk; int ia=1; int seqlength2=seqlength; for(i = 0; i < num_active; i++) // reinitialize all the values to zero { sysinfo[currptr[i]].localpos = 0; } while (1) { for(i = 0; i < num_active; i++) insertion[i] = 0; flaginsert = 0; 103

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for(i = 0; i < num_active; i++) { j = ( int) (100000.0*ran1(idum)); // random number between 0 and 99,999 k = ( int) (3.0*ran1(idum)); // random number between 0 and 2 fprintf(fptest,"%d indelgamma = %lf subgamma = %lf\n",sysinfo[currptr[i]].localpos,indelgam[sysinfo[currptr[i]].localp os],alphagam[sysinfo[currptr[i]].localpos]); if(j < sysinfo[currptr[i]].alpharate alphagam[sysinfo[currptr[i]].localpos]) { if(sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]== 'A') { switch (k) { case 0: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'C'; break; case 1: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'T'; break; case 2: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'G'; break; } } else if(sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]== 'T') { switch (k) { case 0: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'C'; break; case 1: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'A'; break; case 2: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'G'; break; } } else if(sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]== 'C') { switch (k) 104

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{ case 0: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'A'; break; case 1: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'T'; break; case 2: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'G'; break; } } else if(sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]== 'G') { switch (k) { case 0: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'C'; break; case 1: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'T'; break; case 2: sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos]= 'A'; break; } } } // end of subsitution logic l = ( int) (100000.0*ran1(idum)); // random number between 0 and 99,999 if(l < (sysinfo[currptr[i]].indelrate indelgam[sysinfo[currptr[i]].localpos])) { // check if indel will happen d = ( int) (4.0*ran1(idum)); // random number between 0 and 3 rand1 = ( int) (100.0*ran1(idum)); // random number between 0 and 99 // check whether insertion or deletion.. if((rand1 < 50) && (sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] != '-') && (sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] != '\0')) { seqlength++; switch (d) { case 0: 105

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shift(sysinfo[currptr[i]].seq,sysinfo[currptr[i]].localpos); sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] = 'A'; break; case 1: shift(sysinfo[currptr[i]].seq,sysinfo[currptr[i]].localpos); sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] = 'C'; break; case 2: shift(sysinfo[currptr[i]].seq,sysinfo[currptr[i]].localpos); sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] = 'T'; break; case 3: shift(sysinfo[currptr[i]].seq,sysinfo[currptr[i]].localpos); sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] = 'G'; break; default : d = 1; } // end of switch flaginsert = 1; insertion[i] = 1; indelgam = ( double *)realloc(indelgam,(seqlength+1)* sizeof ( double )); alphagam = ( double *)realloc(alphagam,(seqlength+1)* sizeof ( double )); for( int z = seqlength; z > sysinfo[currptr[i]].localpos; z--) { indelgam[z] = indelgam[z-1]; alphagam[z] = alphagam[z-1]; } } // start of the deletion logic else if((rand1 >= 50) && (sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] != '\0')) sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos] = '-'; // inserting a space in the sequence } // end of either insertion or deletion loop sysinfo[currptr[i]].localpos++; 106

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}// end of for loop if(flaginsert == 1) { for(i = 0; i < num_active; i++) { if(insertion[i] == 0) { shift(sysinfo[currptr[i]].seq,sysinfo[currptr[i]].localpos-1); sysinfo[currptr[i]].seq[sysinfo[currptr[i]].localpos-1] = '-'; } sysinfo[currptr[i]].localpos++; } } /* if(seqlength2 < seqlength){ siterate = (double *)realloc(siterate,(seqlength+1)*sizeof(double )); // siterate[seqlength] = gammadev(ia,idum); siterate[seqlength2] = 1.0; seqlength2++; } */ if(sysinfo[currptr[0]].localpos >= seqlength) break; }// end of while loop return ; } /* this function just duplicates the two sequences called from simulate() above */ void branch( int x) { //printf("Entering the branch procedure....\n "); cntr = x+1; while(sysinfo[cntr].seq_id[0] != sysinfo[x].seq_id[0]) cntr++; strcpy(sysinfo[cntr].seq,sysinfo[x].seq); printf("\n\n index = %d, sequence_id = %s.",cntr,sysinfo[cntr].seq_id); sysinfo[cntr].parentbranchtime = sysinfo[x].branchtime + sysinfo[x].parentbranchtime; //make the simulation run for a particular time.. cntr++; strcpy(sysinfo[cntr].seq,sysinfo[x].seq); 107

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printf("\n index = %d, sequence_id = %s.",cntr,sysinfo[cntr].seq_id); sysinfo[cntr].parentbranchtime = sysinfo[x].branchtime + sysinfo[x].parentbranchtime; if(strlen(sysinfo[cntr-1].seq_id) == 1) sysinfo[cntr-1].branchtime = totaltime sysinfo[cntr1].parentbranchtime; if(strlen(sysinfo[2*x+2].seq_id) == 1) sysinfo[cntr].branchtime = totaltime sysinfo[cntr].parentbranchtime; num_active += 2; updatecurrptr(x); // call to function updatecurrptr() no_of_branches += 2; if(no_of_branches == 2*length 2) flag = 1; // length: stores the length of the sequence } /* this function updates the values in the currptr array which points to the number of active sequences * called from branch() above */ void updatecurrptr( int x) { int temp = searchcurrptr(x); int i; currptr[temp] = cntr-1; i = num_active 1; // index starts from 0. while(i>temp + 1) { currptr[i] = currptr[i-1]; i--; } currptr[temp+1] = cntr; } /* function to search for the position of the particular index in the current pointer called from updatecurrptr() above */ int searchcurrptr( int x) { int i; for(i = 0;i<80;i++) { if(currptr[i] == x) return i; 108

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else continue ; } } /* function to shift the indexes of the array called from mutate() above */ void shift( char a[4000], int pos) { int i = pos,len; len = strlen(a); i = len + 1; // a[len+1] = '\0'; while(i>pos) { a[i] = a[i-1]; i--; } } /* this function generates the mask for the given sequence */ void generate_mask() { int i = 0; int a,b,j; b = 10; // upper limit of the random number generator a = 1; // lower limit of the random number generator for(i=0; i<1500; i++) { j = rand() % (b a + 1) + a; // get random number including the max. range mask[i] = j; } } /* function to convert output strings to .pir file called from simulate() above */ void convert( char *s,char *s1) { char s2[4000]; int length, i=0, j=0, flag=0; length = strlen(s); 109

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while(s[i]!='\0') { if(s[i] != '-') { s2[j] = s[i]; j++; } i++; } s2[j] = '\0'; fprintf(fp_pir,">taxon%s\n",s1); // print to .pir fputs(s2,fp_pir); // put sequence in .pir fprintf(fp_pir,"\n"); // print to .pir } 110

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Appendix C: Site Rate Generator Package ################################ # !!! common Makefile !!! # ################################ CPP = c++ OBJECTS1 = siterate.o nrutil1.o sort.o CPPSRCS = siterate.cpp nrutil.cpp sort.cpp all: testrate testrate: $(OBJECTS1) $(CPP) -g -o siterate siterate.o sort.o nrutil1.o siteraete.o: siterate.cpp $(CPP) -g -c -o siterate.o siterate.cpp nrutil1.o: nrutil.cpp $(CPP) -g -c -o nrutil1.o nrutil.cpp sort.o: sort.cpp $(CPP) -g -c -o sort.o sort.cpp siterate: gammadev.o sort.o nrutil.o $(CPP) -o siterate gammadev.o sort.o nrutil.o clean: @echo "'make clean': removing compiled executables and object code" -/bin/rm -f *.o siterate /* siterate.h */ #include #include #include #include #include #include #include double ran1( long *idum); double gamma( double ia, long *idum); double *sort( unsigned long n, double []); /* siterate.cpp */ #include 111

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#include #include #include #include "siterate.h" using namespace std; #define IA 16807 #define IM 2147483647 #define AM (1.0/IM) #define IQ 127773 #define IR 2836 #define NTAB 32 #define NDIV (1+(IM-1)/NTAB) #define EPS 1.2e-7 #define RNMX (1.0-EPS) /* Random number generator */ /* Based on ran1() from Numerical Recipes Returns a random number between 0 and 1. */ double ran1( long *idum) { int j; long k; static long iy=0; static long iv[NTAB]; double temp; if (*idum <=0 || !iy){ if (-(*idum) <1) *idum=1; else *idum = -(*idum); for (j=NTAB+7; j>=0; j--){ k=(*idum)/IQ; *idum=IA*(*idum-k*IQ)-IR*k; if (*idum < 0) *idum += IM; if (j < NTAB)iv[j] = *idum; } iy=iv[0]; } k=(*idum)/IQ; *idum=IA*(*idum-k*IQ)-IR*k; if (*idum < 0) *idum += IM; j=iy/NDIV; iy=iv[j]; iv[j] = *idum; if ((temp=AM*iy) > RNMX) return RNMX; else return temp; } /* Gamma generator */ /* Based on GBH algorithm of Cheng and Feast (Communic. of the ACM. 23, 389-394: 1980). * Generator should not be used for beta <= 0.25 112

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* */ double gamma( double ia, long *idum) { double aa,bb,c,d,t,h1,h2,u,u1,u2,w,tmp; /* if (ia<=0.25) nrerror("Error in gamma: ia <= 0.25 . "); */ aa = ia 0.25; bb = ia/aa; c = 2.0/aa; d = c + 2.0; t = 1.0/sqrt(ia); h1 = (0.4417 + 0.0245*t/ia)*t; h2 = (0.222 0.043*t)*t; tmp = 1.0; while (tmp >= 0.0){ u = ran1(idum); u1 = ran1(idum); u2 = u1+h1*u-h2; /* printf("u = %f, u1 = %f, u2 = %f\n",u,u1,u2); scanf("%d"); */ if(u2 <= 0.0 || u2 >= 1.0) tmp = 1.0; else { /* printf("u2 in the right range");*/ w = bb*pow((u1/u2),4.0); if (c*u2-d+w+1.0/w <= 0.0) return (aa*w); else tmp = c*log(u2)-log(w)+w-1.0; } } return (aa*w); } #undef IA #undef IM #undef AM #undef IQ #undef IR #undef NTAB #undef NDIV #undef EPS #undef RNMX /* siterate.h */ int *ivector( long nl, long nh); void free_ivector( int *v, long nl, long nh); void nrerror( char error_text[]); 113

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/* sort.cpp */ /* Sorts/shuffles the generated site rate variations based on user input */ #define NRANSI #include "nrutil.h" #include "siterate.h" #define SWAP(a,b) temp=(a);(a)=(b);(b)=temp; #define M 7 #define NSTACK 50 double *sort( unsigned long n, double arr[]) { unsigned long i,ir=n,j,k,l=1; int jstack=0,*istack; double a,temp; istack=ivector(1,NSTACK); for (;;) { if (ir-l < M) { for (j=l+1;j<=ir;j++) { a=arr[j]; for (i=j-1;i>=1;i--) { if (arr[i] <= a) break; arr[i+1]=arr[i]; } arr[i+1]=a; } if (jstack == 0) break; ir=istack[jstack--]; l=istack[jstack--]; } else { k=(l+ir) >> 1; SWAP(arr[k],arr[l+1]) if (arr[l+1] > arr[ir]) { SWAP(arr[l+1],arr[ir]) } if (arr[l] > arr[ir]) { SWAP(arr[l],arr[ir]) } if (arr[l+1] > arr[l]) { SWAP(arr[l+1],arr[l]) } i=l+1; j=ir; a=arr[l]; for (;;) { do i++; while (arr[i] < a); do j--; while (arr[j] > a); if (j < i) break; SWAP(arr[i],arr[j]); } arr[l]=arr[j]; arr[j]=a; jstack += 2; if (jstack > NSTACK) nrerror("NSTACK too small in sort."); 114

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if (ir-i+1 >= j-l) { istack[jstack]=ir; istack[jstack-1]=i; ir=j-1; } else { istack[jstack]=j-1; istack[jstack-1]=l; l=i; } } } free_ivector(istack,1,NSTACK); return arr; } #undef M #undef NSTACK #undef SWAP #undef NRANSI /* nrutil.cpp Adapted functions selected from the ANSI C version of the Numerical Recipes utility file nrutil.c */ #include #include #include #define NR_END 1 #define FREE_ARG char* void nrerror( char error_text[]) /* Numerical Recipes standard error handler */ { fprintf(stderr,"Numerical Recipes run-time error...\n"); fprintf(stderr,"%s\n",error_text); fprintf(stderr,"...now exiting to system...\n"); exit(1); } int *ivector( long nl, long nh) /* allocate an int vector with subscript range v[nl..nh] */ { int *v; v=(int *)malloc((size_t) ((nh-nl+1+NR_END)* sizeof ( int))); if (!v) nrerror("allocation failure in ivector()"); return v-nl+NR_END; } void free_ivector( int *v, long nl, long nh) /* free an int vector allocated with ivector() */ { free((FREE_ARG) (v+nl-NR_END)); } 115

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Appendix D: AnchorSearch.cpp //-------------------------header files ------------------------------#include #include #include #include #include #include #include "anchor.h" using namespace std; // ----------------------------------------------------------------------// structure which stores the information about the sequences struct master { char seq[10000]; // stores the sequence char seq_id[100]; // sequence name int length; // stores the length of the sequence }; // declaration of global variables FILE *fp_pir; FILE *fp_rem; FILE *fp_err; char filename[1000]; int cutoff1; int cutoff2; int minLength; master *info; // declaration of subroutines int search_seq( int nseq, char *anchor, int alen); void remove_seq( int n,int nseq); void truncate( int n,int k); void clip( int n,int i); void pir_file( int n); /* Start Of Main */ int AnchorS( char* infile, char* anchor, int forward, int backward, int minLen) { int i = 0; int j = 0; FILE *fp_in; char temp[10000]; char errname[1000]; 116

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char pirname[1000]; char remname[1000]; int nseq = 0; int anchorlen = 0; int test1 = 0; int test = 0; cutoff1 = forward; cutoff2 = backward; minLength = minLen; strcpy(errname,infile); strcat(errname,"_err.txt"); fp_err = fopen(errname,"w"); fprintf(fp_err,"The anchor sequence we are searching for is %s\n\n",anchor); strcpy(filename,infile); strcat(infile,".pir"); fp_in = fopen(infile,"r"); fscanf(fp_in,"%s\n",temp); do{ if (temp[0] == '>') { j++; test1 = fscanf(fp_in,"%s\n",temp); } test1 = fscanf(fp_in,"%s\n",temp); } while (test1 != -1); fclose(fp_in); info = (master *)calloc((j+10), sizeof (master )); // dynamic allocation of struct master fp_in = fopen(infile,"r"); fscanf(fp_in,"%s\n",temp); do{ if (temp[0] == '>') { strcpy(info[i].seq_id,temp); test = fscanf(fp_in,"%s\n",temp); } while (test != -1 && temp[0] != '>') { strcat(info[i].seq,temp); test = fscanf(fp_in,"%s",temp); } info[i].length = strlen(info[i].seq); fprintf(fp_err,"sequence id = %s\n",info[i].seq_id); fprintf(fp_err,"sequence length = %d\n",info[i].length); fprintf(fp_err,"sequence = %s\n\n",info[i].seq); i++; } while (test != -1); nseq = i; fprintf(fp_err,"The value of nseq is %d\n",nseq); 117

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fclose(fp_in); strcpy(pirname,filename); strcpy(remname,pirname); strcat(pirname,"_new.pir"); strcat(remname,"_rem.pir"); fp_pir = fopen(pirname,"w"); fp_rem = fopen(remname,"w"); anchorlen = strlen(anchor); int nseq2 = search_seq(i,anchor,anchorlen); // call to search_seq() which returns number of good sequences fclose(fp_pir); fclose(fp_rem); for (i=0; i
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remove_seq(n,nseq); n--; nseq--; fprintf(fp_err,"\n"); } else{ clip(n,ii); truncate(n,k); int len = strlen(info[n].seq); fprintf(fp_err,"new sequence length = %d\n\n",len); pir_file(n); } } return nseq; } /* function which removes sequences that fail matching criteria and prints them to the rem file called from search_seq() above */ void remove_seq( int n, int nseq) { int q = n; fprintf(fp_rem,"%s\n",info[n].seq_id); fputs(info[n].seq,fp_rem); fprintf(fp_rem,"\n"); while(q < nseq){ info[q] = info[q+1]; q++; } } /* function which shifts the anchor to front of sequence but leaves # of sites in front specified on input called from search_seq() above */ void truncate( int n, int pos) { pos = pos 1 cutoff1; int ll = pos; int len = strlen(info[n].seq); while(ll<=len) { info[n].seq[ll-pos] = info[n].seq[ll]; ll++; } } /* function which cuts off the end of the sequence after the # of points after the anchor specified on input 119

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120 called from search_seq() above */ void clip( int n, int pos) { int q = pos + 1 + cutoff2; int len = strlen(info[n].seq); info[n].seq[q] = info[n].seq[len]; fprintf(fp_err,"\nseq[q] = %c len = %d q = %d\n",info[n].seq[pos+301],len,q); } void pir_file( int n) { fprintf(fp_pir,"%s\n",info[n].seq_id); fputs(info[n].seq,fp_pir); fprintf(fp_pir,"\n"); }

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About the Author Anthony Jason Green was born in Decatur, IL on July 24th, 1979 to John and Dianna Green. He earned his Bachelor of Science in Chemistry magna cum laude from Coastal Carolina University in 2001, wher e his undergraduate research lead to a publication in the journal Inorganic Chemistry. He entered the Doct oral program at the University of South Florida in 2002, and bega n work in computational physical chemistry with Professor Brian Space that Fall. In addition to his research and formal coursework, Mr. Green attended and made presentations at several National Meetings of the American Chemical Society and the Gordon Research Conference on Vibrational Sp ectroscopy. His research has focused on application of the theory and simulation of condensed phase ma tter and spectroscopy toward interesting and important system s found in chemistry, biology, and the environment. Some of the research presente d in this dissertation has been published in the Journal of Chemical Physics and Chemical Physics Letters Additionally, while in the Ph.D. program at the University of Sout h Florida, Mr. Green has served as the web site administrator for the Department of Ch emistry, and has developed an affinity for network administration. He al so has an interest in meteorology, and modeling vortices and other fascinating phenomenon f ound in the atmospheric sciences.


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ABSTRACT: The primary goal of spectroscopy is to obtain molecularly detailed information about the system under study. Sum frequency generation (SFG) vibrational spectroscopy is a nonlinear optical technique that is highly interface specific, and is therefore a powerful tool for understanding interfacial structure and dynamics. SFG is a second order, electronically nonresonant, polarization experiment and is consequently dipole forbidden in isotropic media such as a bulk liquid. Interfaces, however, serve to break the symmetry and produce a signal. Theoretical approximations to vibrational spectra of O-H stretching at aqueous interfaces are constructed using time correlation function (TCF) and instantaneous normal mode (INM) methods. Detailed comparisons of theoretical models and spectra are made with those obtained experimentally in an effort to establish that our molecular dynamics (MD) methods can reliably depict the system of interest. The computational results presented demonstrate the potential of these methods to accurately describe fundamentally important systems on a molecular level.
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