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Construction and application of computationally tractable theories on nonlinear spectroscopy

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Construction and application of computationally tractable theories on nonlinear spectroscopy
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Neipert, Christine L
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Water
Molecular dynamics
Liquid/vapor interface
Nonlinear spectroscopy
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Dissertations, Academic -- Chemistry -- Doctoral -- USF   ( lcsh )
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ABSTRACT: Nonlinear optical processes probe systems in unique manners. The signals obtained from nonlinear spectroscopic experiments are often significantly different than more standard linear techniques, and their intricate nature can make it difficult to interpret the experimental results. Given the complexity of many nonlinear lineshapes, it is to the benefit of both the theoretical and experimental communities to have molecularly detailed computationally amenable theories of nonlinear spectroscopy. Development of such theories, bench marked by careful experimental investigations, have the ability to understand the origins of a given spectroscopic lineshape with atomistic resolution. With this goal in mind, this manuscript details the development of several novel theories of nonlinear surface specific spectroscopies. Spectroscopic responses are described by quantum mechanical quantities. This work shows how well defined classical limits of these expressions can be obtained, and unlike the formal quantum mechanical expressions, the derived expressions comprise a computationally tractable theory. Further, because the developed novel theories have a well defined classical limit, there is a quantum classical correspondence. Thus, semiclassical computational techniques can capture the true physics of the given nonlinear optical process. The semiclassical methodology presented in this manuscript consists of two primary components - classical molecular dynamics and a spectroscopic model. For each theory of nonlinear spectroscopy that is developed, a computational implementation methodology is discussed and/or tested.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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by Christine L. Neipert.
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Construction and Application of Computationally T ractable Theories of Nonlinear Spectroscop y by Christine L. Neipert A dissertation submitted in partial fulllment of the requirements for the de gree of Doctor of Philosoph y Department of Chemistry Colle ge of Arts and Sciences Uni v ersity of South Florida Major Professor: Brian Space, Ph.D. Alfredo Cardenas, Ph.D. Randy Larsen, Ph.D. Lilia W oods, Ph.D. Date of Appro v al: May 23, 2007 K e yw ords: w ater molecular dynamics, liquid/v apor interf ace, nonlinear spectroscop y SFG c r Cop yright 2007, Christine L.Neipert

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Ackno wledgments I w ould lik e to e xpress my gratitude to my husband, Mik e. His lo v e and guidance ha v e been blessings in my life. The unw a v ering support he has of fered and his belief in me has been a true source of jo y and inspiration in my life. I e xtend my deepest and most sincere gratitude, and only hope that my support of him continues to be as meaningful as what he has gi v en to me. My thanks also go out to my current and former group members: Dr Russell DeV ane, Dr Christina Kasprzyk, Dr Angela Perry Dr A. Ben Rone y Mr Jon Belof, Mr T on y Green, Ms. Ashle y Mullen, and Mr Abe Stern. The y ha v e been w onderful friends and colleagues. Man y thanks also to Professor Preston Moore, whose help and advice has been in v aluable. I w ould also lik e to thank my committee: Dr Alfredo Cardenas, Dr Randy Larsen, and Dr Lilia W oods. Finally o v er the past four years, I ha v e been guided and mentored by a remarkable person, Professor Brian Space. In e v eryone' s life, there are those fe w people who hold special signicance; Professor Space has opened up a w orld of opportunity to me, and has been one of the most positi v e inuences in my life. W ords or deeds cannot possibly be gin to e xpress my gratitude, and the deep respect that I ha v e for him. I of fer my humblest thank you.

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Note to Reader Note to Reader: The original of this document contains color that is necessary for understanding the data. The original dissertation is on le with the USF library in T ampa, Florida.

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T able of Contents Abstract v 1 Introduction 1 2 Theoretical Background 3 2.1 Generalized Density Matrix Theory Approach 3 2.2 T ime Correlation Descriptions & Implementation Considerations 10 2.3 A Basic Spectroscopic Model 12 3 Second Order Surf ace Specic Sum V ibrational Frequenc y Spectroscop y (SVFS) 16 3.1 Theoretical De v elopment of the Go v erning SVFS TCF 18 3.2 SVFS – Application of Theory & Computational Methods 26 3.3 Concluding Remarks on Ne w W ater Model & Future Impro v ements 39 4 Quadrupole Induced Bulk SVFS 40 4.1 General Theoretical De v elopment of Quadrupole Contrib utions 43 4.2 TCF Expressions for SVFS Quadrupolar Susceptibilities 48 4.3 Calculation via a Char ge-Interaction Model 51 5 Static Field Induced Third Order SVFS 55 5.1 Ef fecti v e Polarization Due to A Static Field in Isotropic Media 56 5.2 Microscopic (3) Expression to Account for a Static Field 58 5.3 A TCF Approach to Quantify (3) ;R E S Contrib utions 62 5.4 Methods of Computational Implementation 70 i

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6 Linear Raman: A Frequenc y-T ime Deri v ation of the Response Function 78 7 Conclusion 82 References 84 About the Author ii End Page

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List of Figur es 3.1 Experimental SVFS Spectra of the W ater/V apor Interf ace 33 3.2 Impro v ed SVFS Spectra of the W ater/V apor Interf ace 34 3.3 SVFS Spectra of the W ater/V apor Interf ace 35 4.1 The Detection Directions of Second Order Optical Ef fects 42 iii

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List of T ables 2.1 A Summary of Second Order Optical Processes 9 3.1 Experimental Equilibrium Conguration of Gaseous W ater 29 3.2 Equilibrium Conguration of an MD W ater Model 30 3.3 Gaseous Polarizability T ensor of a W ater Molecule 30 3.4 Gas Phase Polarizability Deri v ati v e T ensor of W ater 31 5.1 Hyperpolarizability T ensor P arameters for W ater and Results Produced 73 5.2 Hyperpolarizability Deri v ati v e P arameters for the NSM of Gaseous W ater 74 5.3 Gaseous W ater Interaction Hyperpolarizability Deri v ati v e T ensor with NSM 76 i v

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Construction and A pplication of Computationally T ractable Theories of Nonlinear Spectr oscopy Christine L. Neipert ABSTRA CT Nonlinear optical processes probe systems in unique manners. The signals obtained from nonlinear spectroscopic e xperiments are often signicantly dif ferent than more standard linear techniques, and their intricate nature can mak e it dif cult to interpret the e xperimental results. Gi v en the comple xity of man y nonlinear lineshapes, it is to the benet of both the theoretical and e xperimental communities to ha v e molecularly detailed computationally amenable theories of nonlinear spectroscop y De v elopment of such theories, bench mark ed by careful e xperimental in v estig ations, ha v e the ability to understand the origins of a gi v en spectroscopic lineshape with atomistic resolution. W ith this goal in mind, this manuscript details the de v elopment of se v eral no v el theories of nonlinear sur f ace specic spectroscopies. Spectroscopic responses are described by quantum mec hanical quantities. This w ork sho ws ho w well dened classical limits of these e xpressions can be obtai ned, and unlik e the formal quantum mechanical e xpressions, the deri v ed e xpressions comprise a computationally tractab le theory Further because the de v eloped no v el theories ha v e a well dened classical limit, there is a quantum classical correspondence. Thus, semiclassical computational techniques can capture the true ph ysics of the gi v en nonlinear optical prov

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cess. The semiclassical methodology presented in this manuscript consists of tw o primary components – classical molecular dynamics and a spectroscopic model. F or each theory of nonlinear spectroscop y that is de v eloped, a computational implementation methodology is discussed and/or tested. vi

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Chapter 1 Intr oduction The interaction of light and matter Seemingly simple, the true comple xity of thi s inter action w as be ginning to be realized as early as 300 B.C. with the w ork of Euclid. Names such as K epler Ne wton, Maxwell, and Einstein dot the landscape in t he e v olution of this scientic frontier [1, 2] The culmination of this w ork lead to a signicant technological inno v ation in the mid-twentieth century: the de v elopment of the rst contemporary laser and in turn, the eld of modern optical spectroscop y [2–5] Modern optical spectroscopi c techniques are po werful tools for probing the structure and dynamics of molecular systems. [2–16] This is well i llustrated by comparing a gi v en syste m' s g as and condensed phase spectra. Condensed phase spectra can sho w e .g shifts in fundamental resonant frequencies in comparison to their g as phase spectra, and/or e xhibit such ef fects as motional narro wing. The dif ferences between the spectra of the tw o phases are due to the man y-body ef fects that are present in the condensed phase and that are absent in the g as phase. Further studying spectroscop y from a theoretical perspecti v e is enticing because the underlying molecular interactions probed by a specic type of spectroscop y can be realized with proper analysis and comparison to e xperiment. [6, 17–19] Be ginning with the pioneering, and no w f amous, w ork of Bloember gen, [20–23] Gor don, [24–29] and man y others in the mid-twentieth century talented scientists ha v e been 1

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constructing theories of linear and nonlinear spectroscop y This body of w ork continues on in that tradition, b ut using a ne w unique, approach which is amenable to semiclassical computation techniques. In Chapter 2, the general theory of spec troscop y is presented and discussed. In subsequent chapters, theories which use the gene ral formalism, detailed in Chapter 2, as the starting point in their construction are presented and di scussed – these include: dipolar sum vibrational frequenc y spectroscop y (SVFS), multi-polar SVFS, and third order SVFS. In the nal chapter preceding the concluding remarks, the no v el approach used to describe v arious SVFS processes is applied to de v elop the well kno wn Optical K err Ef fect (OKE) result. This is done to mak e a connection wit h the more common Loui ville space approach, and e xpli citly e xpress the implicit assumptions that are present in these other techniques. 2

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Chapter 2 Theor etical Backgr ound 2.1 Generalized Density Matrix Theory A ppr oach In typical electric optical laboratory e xperiments, a light eld is impinged on a system, and the resulting polarization, P is measured. [2] Letting M denote the number of system molecules, the system dipole moment operator and a set of angle brack ets a quantum mechanical e xpectation v alue the relationship between the measured polarization and system dipole moment operator is gi v en by: P = M h i = M h i (2.1) In the rst part of this equal ity the bar indicates the quantum mechanical e xpectation v alue is to be further ensemble a v eraged. This secondary a v erage is tak en because of our inability to determine the e xact man y-body w a v efunction of the system as it e v olv es in time. [2, 4–6] (This inability is a pure cla ssical statistical phenomenon as opposed t o an inherent uncertainty due to quantum mechanical constraints.) In the second portion of the equality in Equation 2.1, the density matrix, has been incorporated to account for this statistical uncertainty F ormally the density matrix between states n and m is gi v en by: mn = X s p ( s ) a ( s ) m ( t ) a ( s ) n ( t ) (2.2) 3

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Here, p ( s ) denotes the probably of the system being in state s and the a ( s ) coef cients are time dependent probability amplitudes in standard f ashion. In the limit the e xact state of the w a v efunction were kno wn, the distrib ution p ( s ) w ould reduce t o a delta function, and Equation 2.1 w ould simplify to a typical non-ensemble a v eraged e xpectation v alue. As a system e v olv es in time under the inuence of a perturbation, the w a v efunction changes. This e v olution is formally described by the time dependent Schr ¨ odinger equation. [30] Inclusion of the classical stat istical uncertainty in this equation results in an equation of motion for the density matrix: d nm dt = i ~ [ ; H T ] nm (2.3) Here, H T denotes the total man y-body Hamiltonian of the system. It is common to further partition Equation 2.3 into tw o pieces to f acilitate theoretical de v elopment. d nm dt = i ~ [ ; H ] nm r nm ( nm ( eq ) nm ) (2.4) In Equation 2.4, interactions, such as atomic collisions, that are not easily mathematically incorporated into the Hamiltonian and contrib ute to the system' s decay are separated out. [2, 4] These interactions are represented by the second term in Equation 2.4. Here, r nm = r mn and is a phenomenological, purely real, damping f actor that weights ho w quickly the system will relax back to the equilibrium density matrix v alue when, e .g an atomic collision occ urs. Generally this equation of motion cannot be solv ed analytically b ut can be e xpanded in a po wer series and solv ed order by order T o proceed in the deri v ation, it is further assumed that: (1) the Hamil tonian, H is separable into equilibrium and perturbed components ( H = H ( o ) + H (1) ), and (2) (0)nm = ( eq ) nm n = m 0 n 6 = m In essence, (2) constrains the proscribed mathematical descri ption to a system in which an y state that is occupie d in equilibrium must be in a population, and fur ther requires that an y e xcitation does not produce a coherent superposition of states. [2] 4

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Proceeding with an ordered e xpansion of the density m atrix such that a solution can be used to solv e for the primary quantity of interest – the polarization (Equation 2.1): d nm dt = X i d ( i ) nm dt = i ~ X i [ ( i 1) ; H (1) ] nm E nm ( i ) nm ir nm ( ( i ) nm ( eq ) nm ) (2.5) Here, E nm is dened as E n E m and i s the ener gy dif ference between states n and m Also, note that for i = 0 the commutator in Equation 2.5 must be zero because the z eroth order density matrix describes the system in equilibrium – i.e prior to the perturbation, H (1) Solving the density matrix equation of motion in a perturbati v e manner f acilitates writing the observ ed polarization in terms of an ordered e xpansion also: P = X N P ( N ) = M X N =0 h ( N ) i (2.6) Equation 2.6 can pro vide further ph ysical insight by considering that the specic phase and intensity of the detected polarization will depend on the intrinsic response function of the system as well as the properties of the applied perturbing eld(s). By denition, a system' s intrinsic response function pro vides a complete description of it under all conditions. (The label of response function is generally used to describe this quantity in the time domain. Ho we v er in the frequenc y domain, the response is typically referred to as the susceptibility .) In pursuit of this ph ysical insight, consider a general perturbing electric eld (that may or may not be the only perturbing eld) applied at time t and position r : E ( r ; t ) = N F X n E n ( t ) e i k n r + E n ( t ) e i k n r (2.7) In Equati on 2.7, k n is t he w a v e v ector specifying the electric eld' s propag ation direction, and components that are slo wly v arying in space and those that are spatially highly oscillatory ha v e been partitioned. [3, 5, 31] The slo wly v arying spacial component, E n ( t ) 5

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can generally be further decom posed into temporally ( n ( t )) and spatially ( E n ) dependent parts. [31] This subsequent separation allo ws the eld to be re written in the form: E ( r ; t ) = N F X n E n n ( t ) e i k n r + E n n ( t ) e i k n r (2.8) In Equations 2.7 and 2.8, the sum on n ranges o v er the number of applied perturbing elds ( N F ), and is included because, in the most general case, e xact time ordering of multiple appli ed elds cannot be assumed. [5] In practice, e xperiments in the time domain typically use relati v ely short pulses that are separated and ordered in time while the frequenc y domain techniques emplo y nearly monochromatic laser elds that o v erlap in time and space – such considerations simplify the required analysis considerably Gi v en the denition of the eld in Equation 2.8, the most general description of the N th component of the polarization (Equation 2.6), tak es the form of a multiple time inte gration o v er the N th order material response function, R ( N ) : P ( N ) ( r ; t ) = Z 1 0 d 1 Z 1 0 d N R ( N ) ( 1 ; ; N ) j E ( r ; t 1 ) E ( r ; t N ) (2.9) Here, R ( N ) is an ( N + 1) rank ed tensor and the j represents N tensor contractions. In this re write of the polarization, it should not be o v erlook ed that R ( N ) is a quantum mechanical object which is order ~ N dependent and is a function of the perturbed Hamiltonian operator In Equation 2.9, the time inte grations are necessary to account for the f act that there will be a time delay between application of the eld(s) and the response of the system. [6, 32, 33] If both processes were simultaneous, the inte gra tions w ould not be necessary This simultaneous eld application/system response limit is commonly referred to as the delta function pulse limit, and while it is not strictly e v er true, it is often assumed from a theoretical standpoint to simplify calculation of the polarization. [5] Unlik e the delta function pulse limit, the analogous limit in frequenc y space is not unph ysical because while a true delta function pulse is not achie v able, a nearly monochromatic eld 6

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is. [3, 5] In the limit of monochromatic elds, E ( i ) = 2 E i ( i n i ) and the N th order total polarization e xpressed as a frequenc y n is gi v en by a simple equality: [6] P ( N ) ( r ; n) = (2 ) N 1 ( N ) ( 1 :::! N ) j E 1 ( 1 n 1 ) ::: E N ( N n N ) (2.10) Equation 2.10 describes the total N th order polarization a gi v en sample emits when optically perturbed by N total interactions with a single or se v eral monochromatic electric elds. Ho we v er it is not necessarily common to measure the total N th order polarization because there are unique optical processes that contrib ute to the total N th order polar ization, and can be detected independently This later method is adv antageous because more information re g arding the sys tem can be realized by analyzing each contrib ution to the total polarization independently Specically insertion of Equation 2.8 into Equation 2.9 implies there are (2 N F ) N F terms presen t for each P ( N ) ( r ; n) component. Each of the (2 N F ) N F terms represent a distinct optical process that is ra diated in a specic direction and frequenc y determined by combinations of the incident elds' w a v e v ectors and frequencies. (Note, a eld' s frequenc y and w a v e v ector are directly related, and by conv ention, possess the same sign.) F or e xample, consider tw o perturbing el ds where the response is not instantaneous. From Equation 2.8, and letting c:c : represent a gi v en term' s comple x conjug ate, it follo ws that the elds are formally e xpressed as: E ( r ; t 1 ) = E 1 ( t 1 ) e i k 1 r + c:c : + E 2 ( t 1 ) e i k 2 r + c:c : (2.11) E ( r ; t 2 ) = E 1 ( t 2 ) e i k 1 r + c:c + E 2 ( t 2 ) e i k 2 r + c:c : (2.12) If the product of these tw o elds is tak en (Equation 2.6), 16 terms result. These are summarized in T able 2.1. T able 2.1 also serv es to further reinforce that specic combinations of the applied elds' w a v e v ectors will result in unique polarization signals 7

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which are emitted from the sample in highly specic directions. The detected signal from a specic optical process detected a frequenc y s and w a v e v ector k s is denoted by P ( N ) ( k s ; s ) and the sum of all the polarization signals (16 in this case) is the total polarization, P ( N ) ( r ; n) No w consider a single perturbing eld that is not suf ciently intense in the sense that it w ould not in v alidate the perturbati v e description that has been de v eloped thus f ar In this case – ne glecting the zeroth order term from here on because it is a constant corresponding to an unperturbed system – the series in Equation 2.6 (or equi v alently Equation 2.9) could reasonably be truncated at rst order Progressing this ar gument, suppose that tw o elds were applied to the system, then the leading term detected in all k i + k j directions (where i; j = 1 or 2 ) w ould be dominated by the corresponding second order term of Equa tion 2.6. This line of reasoning can be e xtended to describe N F applied elds, and it v ery useful for g aining ph ysical insight into an N th order optical spectroscopic process. This is because, within the outlined formalism, the ge neral e xpression for the system' s response to an optical process in v olving N F applied elds and detect ed in a highly specic direction can be found via solving for the corresponding N th order density matrix (Equation 2.5). It is upon this foundation that computationally amenable theoretical descriptions of specic optical processes are de v eloped from in later chapters of this manuscript. Note, as opposed to introducing the density matrix, the man y-body w a v efunction could be solv ed in a perturbati v e nature. Ho we v er the density matri x pro vides a more ph ysically insightful description of the system because populations and intermediate co8

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E 1 ( t 1 ) e k 1 r E 1 ( t 2 ) e i k 1 r k s = 2 k 1 s = 2 1 E 1 ( t 1 ) e k 1 r E 1 ( t 2 ) e i k 1 r k s = 0 s = 0 E 1 ( t 1 ) e k 1 r E 2 ( t 2 ) e i k 2 r k s = k 1 + k 2 s = 1 + 2 E 1 ( t 1 ) e k 1 r E 2 ( t 2 ) e i k 2 r k s = k 1 k 2 s = 1 2 E 1 ( t 1 ) e i k 1 r E 1 ( t 2 ) e i k 1 r k s = 0 s = 0 E 1 ( t 1 ) e i k 1 r E 1 ( t 2 ) e i k 1 r k s = 2 k 1 s = 2 1 E 1 ( t 1 ) e i k 1 r E 2 ( t 2 ) e i k 2 r k s = k 1 + k 2 s = 1 + 2 E 1 ( t 1 ) e i k 1 r E 2 ( t 2 ) e i k 2 r k s = k 1 k 2 s = 1 2 E 2 ( t 1 ) e i k 2 r E 1 ( t 2 ) e i k 1 r k s = k 1 + k 2 s = 1 + 2 E 2 ( t 1 ) e i k 2 r E 1 ( t 2 ) e i k 1 r k s = k 1 + k 2 s = 1 + 2 E 2 ( t 1 ) e i k 2 r E 2 ( t 2 ) e i k 2 r k s = 2 k 2 s = 2 2 E 2 ( t 1 ) e i k 2 r E 2 ( t 2 ) e i k 2 r k s = 0 s = 0 E 2 ( t 1 ) e i k 2 r E 1 ( t 2 ) e i k 1 r k s = k 1 k 2 s = 1 2 E 2 ( t 1 ) e i k 2 r E 1 ( t 2 ) e i k 1 r k s = k 1 k 2 s = 1 2 E 2 ( t 1 ) e i k 2 r E 2 ( t 2 ) e i k 2 r k s = 0 s = 0 E 2 ( t 1 ) e i k 2 r E 2 ( t 2 ) e i k 2 r k s = 2 k 2 s = 2 2 T able 2.1: Column 1 details the sixteen terms resulting from tw o elds incident on a sample. Columns 2 and 3 gi v e the corresponding signal w a v e v ector k s and signal frequenc y s that a specic optical process is detected at. Notice that some processes ha v e multiple contrib utions, and a term al w ays has a comple x conjug ate such that the total polarization is al w ays real. 9

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herences between states become mathematically ob vious – whereas only be ginning and ending populations are mathematically ob vious in the correction to the w a v efunction for malism. [5, 34] 2.2 T ime Corr elation Descriptions & Implementation Considerations As will be demonstrated in later chapters of this manuscript, the N th order response (susceptibility) a gi v en e xperiment detects can be e xpressed in terms of sums and dif ferences of time correlation functions (TCF' s). [5, 6, 35, 36] F or e xample, it can be sho wn [5, 37] that the linear response component emitted from a standard FTIR e xperiment is proportional to the autocorrelation function of the system' s dipole operator (Equation 2.13). (T raditional frequenc y domain spectra, in the limit of monochromatic elds, are obtained via F ourier transform of Equation 2.13.) R ij (1) ( k s ; t ) / i ~ Z 1 0 dt 0 h i (0 ) j ( t 0 ) i h i ( t 0 ) j (0 ) i (2.13) The TCF' s in Equation 2.13 are comple x quantum mechani cal objects – which forces the question: Ho w can these quantities be amenable to classical or semiclassical computation techniques? Before addressing this question, it is instructi v e to e xpand the comple x TCF' s in terms or their real and imaginary parts. Not ing that h i (0 ) j ( t 0 ) i = fh i ( t 0 ) j (0 ) ig and letting C I ( t ) [ C R ( t ) ] denote the imagina ry [real] component of the TCF Equation 2.13 can analytically be re written as: R ij (1) ( k s ; t ) / 2 ~ Z 1 0 dt 0 C I ( t 0 ) (2.14) 10

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Further representing the TCF' s in Equation 2.13 in their Heisenber g representation and independently F ourier transforming them, re v eals a detailed-balance relationship between the real and imaginary parts: C I ( ) = tanh ( ~ ) C R ( ) [5, 6] Substitution of this relationship into Equation 2.14 and T aylor e xpansion of the tanh f actor reduces the necessary order of ~ and therefore, establishes a deniti v e classical limit. (The e xplicit mathematical deri v ation of the detailed-balance relationship will be presented in Chapter 3.) In this limit, the system' s response becomes proportional to the purely real autocorrelation function of the system' s dipole – a quantity that is amenable to standard molecular dynamics (MD) simulation techniques. [6, 17–19, 36, 38, 39] In principle, it is an achie v able goal to incorporate all the necessary potential terms (parametrized with ab initio calculations and/or e xperimental data) in an MD simulation to accurately reproduce the system' s electric moments that are required to compute TCF' s that correspond to a specic response of a gi v en type of spectroscop y [40, 41] Ho we v er consider e .g Equation 2.13 and the McLaurian e xpansion of the system' s dipole operator around position r : ab = o ab + 0 r ab + O ( r 2 ) The spectroscopic signal a gi v en FTIR e xperiment measures stems from the perturbing eld inducing a transition between states ( i.e a 6 = b ). Therefore R (1) ij in Equation 2.13 must be proportional to the rst deri v ati v e of the dipole operator Ag ain, in principle it is possible to accurately capture this quantity using a classical MD potential parametrized with e xperimental data or ab initio calculations when there is a lac k of e xperimental data. [40, 41] Ho we v er in practice, it is much more straightforw ard to use trajectories generated from a classical, comparably 11

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simple, MD potential that models the system' s a v erage electric moments accurately b ut does not necessarily capture the true v alue of the deri v ati v es of these quantities. It is v ery important to note that this type of MD simulation will still capture the true dynamics of the system; the system' s electric moment deri v ati v es correspond to innitesimal changes in an atom' s local en vironment, and therefore do not play an y signicant role in go v erning the dynamics of the system. [6, 42] It is the electric moment deri v ati v es, ho we v er that provide a windo w into the comple x ph ysics of the system via spectroscop y Herein, we ha v e de v eloped a “spectroscopic model” that uses the trajectories generated from a “simple” MD potential as input, and allo ws for the system' s a v erage electric moments and their deri v ati v es to be calculated for each time step along the trajectory In the ne xt section, the foundations of the basic spectroscopic model will be outlined. 2.3 A Basic Spectr oscopic Model A spectroscopic model – bas ed upon a Thole-Applequist interaction model [43–46] – has been constructed. The underlying task of the spectroscopic model is to capture ho w the electrostatic proper ties of one atom inuences the electric moments of another atom, and ultimately ho w the atoms collecti v ely interact to produce the system' s total electrostatic moments. In pursuit of de v eloping this mathematical model, con sider a system of a single molecule that is itself composed of se v eral atoms. There is some intrinsic internal eld v ector F i at each atom ”i” due to all the other atom' s in the molecule. If an e xternal eld is no w applied to this molecule, the e xternally applied eld will ha v e a local v alue of E ( o ) i 12

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at atom ”i”. Further the presence of the applied eld can induce a dipole at each atom ”i”, and the induced dipole on atom ”i” can in turn induce a dipole on atom ”j”. No w if the total dipole at each atom can be e xpanded in a McLaurian series around the total electric eld, E ( to t ) i it can be writte n as Equation 2.15. Note, all Roman subscripts denote atoms, and are not to be summed o v er unless s pecied. All Greek superscripts denote Cartesian tensor components, and are subject to the Einstein summation con v ention. mo l = X i i = X i n ( o ) ; i + 0 i E ( to t ) ; i + O [( E ( to t ) i ) 2 ] o (2.15) Here, ( o ) ; i is the intrinsic dipole of atom ”i” (intrinsic because the e xpansion is a McLaurian series around the total electric eld). Note the second rank ed tensor 0 i is generally denoted by i and is the intrinsic polarizability of atom ”i”. The polarizability of a molecule is a measure of ho w easily a dipole can be induc ed. Also notice Equation 2.15 has been truncated at rst order The second and third deri v ati v es of the dipole e v aluated at zero total eld are often referred to as an atom' s intrinsic h yperpolarizability and second h yperpolarizability respecti v ely T o f acilitate mathematical de v elopment of the spectroscopic m odel, the total electric eld will be split into its three distinct contrib utions: E ( o ) i + F i + ( P j T ij j = P j 5 5 r 1 ij j ). The third term corresponds to the eld at atom ”i ” resulting from induced moments on all other atoms. [44, 47] mo l = X i i = X i ( ( o ) ; i + 0 i ( E ( o ) ; i + F i + X j 6 = i T r ij rj ] )) (2.16) The deri v ati v e of the total dipole e v aluat ed at zero total eld strength has already been identied as the intrinsic system polarizability Therefore, it is instructi v e to wri te the 13

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partial deri v ati v e of Equation 2.16 with respect to the applied eld, E ( o ) i because, this can be associated with the system polarizability under the inuence of a static electric eld. A mo l = @ mo l @ E ( o ) ; = X i @ i @ E ( o ) ; k = X i i ( ik + X j 6 = i T r ij @ rj @ E ( o ) ; k ) (2.17) A j k = i j k ij r i P j T ij (2.18) Equation 2.18 is a primary result of this section because it pro vides a mathematical means to calculate the man y-body polar izability matrix, A Further Equation 2.18 allo ws the total induced system dipole, in d; mo l to be e xpressed as: in d; mo l = A mo l ( E ( o ) ; + F ) (2.19) Note, whi le this present theoretical discussion and mathematical de v elopment w as conducted under the assumption that the system w as composed of a single molecule, this w as done for conceptual clarity only The equations de v eloped in thi s section, as is, generalize to a system contain an y number of molecules. T ranslation between this gener al model de v eloped and our spectroscopic model inv olv es: (1) assuming the atomic polarizability tensors can be approximated as point polar izabilities instead of tensors, i i I (2) ass uming a constant applied eld, and (3) introduction of a dampin g f actor to pre v ent unph ysically lar ge induced dipole moments. (1) and (2) are not necessary and serv e only to simply computational ef forts. In contrast, (3) is necessary because the system' s induced dipole, is a function of inter -nuclear distances, and when an interatomic distance, r ij approaches (4 i j ) 1 = 6 unph ysically lar ge 14

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v alues for the induced-dipole moments result. [43] In follo wing chapters, the specics of ho w the interacti on model is parametrized for a gi v en system will be outlined in detail. F ollo wing, for the sak e of completeness and clarity these basic steps will be briey summarized. W ithin the spectroscopic model, an y general system is rst brok en do wn into the types of molecules that comprise it, and then into the atoms that comprise the specic sets of the dif ferent molecular species. Each atom type within a lik e group of molecular species is assigned a point char ge (to reproduce the permenant dipole) and polarizability along with point deri v ati v es. The v alue of the zeroth and rst order terms in the McLaurian e xpansion for the dipole and polarizability moments (as detailed in the pre vious section) are chosen such that when the atoms comprising a particular isolated molecular species are allo wed to interact, the molecule' s electric moments and their deri v ati v es are accurately reproduced. In tting the model, the zerot h order terms are parametrized rst, and are based upon a molecules equilibrium g as phase conguration. The rst order terms are subsequently parametrized, and are based upon performing man y calculations within our spectroscopic model where small displacements ( O ( E 5 A ) ) from the equilibrium conguration are made such that point dif ference deri v ati v es can be obtained. Note, the molecule' s g as phase electric moments and their deri v ati v es are t to e xperimental data and/or ab initio calculations. 15

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Chapter 3 Second Order Surface Specic Sum V ibrational Fr equency Spectr oscopy (SVFS) The possible second order optical process that occur when a system is perturbed by tw o impinging electric elds are Sum Harmonic/Sum Frequenc y/Sum Dif ference Generation (SFG/SHG/DFG) and optical rectication. Sum vibration frequenc y spectroscop y (SVFS) is a vibrationally resonant v ersion of SFG, and is a po werful e xperimental method for probing the structure and dynamics of interf aces. SVFS e xperiments typically emplo y tw o elds, a visible (vis ) and an infrared (IR), o v erlapped in time and space at an inter f ace. In most cases, the frequenc y of the visible eld is x ed, and the IR frequenc y range is scanned. The signal SVFS measures is proportional to the second order polarization component in the sum w a v e v ector direction and at the sum frequenc y of applied perturbing elds: P (2) ( k s = k v is + k I R ; s = v is + I R ) In the absence of an y vi brational resonance at the instantaneous IR laser frequenc y a structureless signal due to the static h yperpolar izability of the interf ace is obtained. [48–50] When the IR laser frequenc y is in tune with a vibration at the interf ace, a resonant lineshape ha ving a characteristic shape that reects both the structural and dynamical en vironment at the interf ace, is produced. [51–53] 16

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SVFS, as well as all e v en ordered polarization measurements, are interf ace specic in the dipole approximation due to symmetry constraints. This can be understood by considering an isotro pic system with an y number of elds applied to it. If the direction of all the electric elds in an e xperiment were re v ersed, the sign of The polarization must change because all directions are equi v alent on a v erage. [2] Ho we v er e v en numbers of applied elds will mak e the polarization equal to its ne g ati v e – a condition that insists the polar ization is zero, i.e P = P = 0 [5] At an interf ace, or in certain noncentrosymmetric solids, [3] the isotrop y of the system is brok en. This leads to a second order signal within the dipole approximation, and in this case, the signal is proportional to the product of the susceptibility and the electric elds as described by Equation 2.10. Recent years ha v e seen a great increase in the number of e xperimental groups per forming SVFS in v estig ations. [7, 12, 54–61] In contrast, molecularly detailed theoretical simulations of SVFS spectra are comparati v ely fe w and ha v e only recently be gun making a signicant impact. Lik e all vibrational spectroscopies, the goal of SVFS is to infer structural and dynamical properties from the observ ed spectroscopic signatures. In contrast to more traditional vibrational spectroscopies, SVFS lineshapes tend to be more comple x (reecting the unique en vironment that is present at an interf acial boundary), and are not nearly as well understood. Thus, the adv ent of ef fecti v e theoretical simulation techniques promises to help realize the potential of SVFS to permit detailed characterization of inter f aces on par with that done in the b ulk. T o this en d, a TCF theory of SVFS, based upon the densit y matrix formalis m outlined in Chapter 2, and amenable to semiclassical com17

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putational t echniques w as de v eloped. F ollo wing, the construction of this theory will be outlined and discussed. Subsequently its practical application to a liquid w ater interf ace will be presented. 3.1 Theor etical De v elopment of the Go v er ning SVFS TCF F ormally a second order polarization, the dominant r space contrib ution in a typical SVFS e xperiment is gi v en by: P (2) ( r ; I R ; v is ) = h (2) i = 2 X k s e i ( k s ) r (2) ( I R ; v is ): E 1 E 2 (3.1) Equation 3.1 assumes the dipole approximation. Therefore, the perturbing Hamiltonian operator is dened as: H (1) ( t ) = E ( t ) From a theoretical standpoint, the primary quantity of interest is the system' s susceptibility tensor , because whereas the elds are input v ariables, the susceptibility is an intrinsi c property of the system, and contains all the information that can be probed. [2–6] A microscopic e xpression for (2) can be de v eloped by solving for the second order density matrix, and equating the latter tw o equalities in Equation 3.1. Gi v en the perturbati v e approach outlined in Chapter 2, the solution to the second or der density matrix can be seen to be a function of the rst order density matrix, and the solution to the rst order density matrix can be seen to be a function of the zeroth order density matrix. The zeroth order densi ty matrix is kno wn, and is gi v en by a standard Boltzmann distrib ution: (0) = eq = e H Q where Q is the partition function and is the 18

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reciprocal of Boltzmann' s const ant multiplied by temperature. The solution for (1) under the stated conditions will be found rst. (1)nm ( t ) = i ~ e it ( nm ir nm ) Z t 1 dt 0 [ H (1) ( t 0 ) ; (0) ] nm e it 0 ( nm ir nm ) (3.2) Isolating the commutator in Equation 3.2, and projecting with a complete set of states, v results in: [ H (1) ( t ) ; (0) ] nm = X v nv (0)v m E (t) + (0)nv v m E (t) (3.3) As discussed in Chapter 2, the zeroth order density matrix is assumed to be diagonal. This approximation allo ws Equation 3.3 to be written as Equation 3.4, and this representation of the commutator can be substituted back into Equation 3.2 to gi v e Equation 3.5. [ H (1) ( t ) ; (0) ] nm = ( (0)nn nm nm (0)mm ) E (t) (3.4) (1)nm ( t ) = i ~ e it ( nm ir nm ) ( (0)mm (0)mm ) nm X p E ( p ) Z t 1 dt 0 e it ( p nm + ir nm ) (3.5) Note in Equation 3.5, the applied e ld has been represented in terms of a F ourier series, and p is a completely ge neral frequenc y at this point. Analytic ( R t 1 dt 0 e it 0 ( a ib ) = i a ib e it ( a ib ) ) inte gration of Equation 3.5 yields: (1)nm ( t ) = 1 ~ ( (0)mm (0)nn ) X p e it p nm E ( p ) nm p ir nm (3.6) Equation 3.6 is the perturbati v e solution to the rst order po wer series e xpansion of the density matrix equation of motion. It is from this term that an e xpression for the linear 19

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susceptibility (1) can be determined, and it i s also the starting point for solving for the second order density matrix. W e proceed by solving for (2) (2)nm ( t ) = i ~ e it ( nm ir nm ) Z t 1 dt 0 [ H (1) ( t 0 ) ; (1) ( t )] nm e it 0 ( nm ir nm ) (3.7) Expanding the commutator and representing the second applied eld in terms of a F ourier series that is a function of a general q gi v es: [ H (1) ( t ) ; (1) ( t )] nm = X v [ (1)nv v m nv (1)v m ] E (t) = 1 ~ X v pq (0)nn (0)v v p nv + ir nv [ nv E ( q )][ v m E ( q ))] e it ( p + q ) 1 ~ X v pq (0)v v (0)mm p v m + ir v m [ v m E ( p )][ nv E ( q )] e it ( p + q ) (3.8) Inserting the e xpanded commutator in Equation 3.8 into Equation 3.7, and subsequently performing the necessary inte gration yields: (2)nm ( t ) = 1 ~ 2 X v pq e it ( p + q ) ( (0)nn (0)v v )[ nv E ( p )][ v m E ( q )] ( p nv + ir nv )( p + q nm + ir nm ) e it ( p + q ) ( (0)v v (0)mm )[ v m E ( p )][ nv E ( q )] ( p v m + ir v m )( p + q nm + ir nm ) # (3.9) Making a change of dummy indices in Equation 3.9 and substituting into Equation 3.1 produces a dipolar general e xpression for the second order susceptibility Building in intrinsic permutation symm etry [2] to the second order susceptibility and no w specify that p = v is q = I R and I R + v is = s results in an e xpression for (2) that is a sum of eight terms. These eight terms a re detailed belo w Note, Equation 3.10 e xpresses 20

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the total second order susceptibility a third rank ed tensor in terms of its Cartesian tensor components. (2)ij k = M ~ 2 X mng (0)g g ig n jnm kmg ( s ng + ir ng )( I R mg + ir mg ) jg n inm kmg ( v is + ng + ir ng )( I R mg + ir mg ) + kg n jnm img ( s + mg + ir mg )( I R + ng + ir ng ) kg n inm jmg ( v is mg + ir mg )( I R + ng + ir ng ) + jg n img knm ( s + mg + ir mg )( v is + ng + ir ng ) + ig n jmg knm ( s ng + ir ng )( v is mg + ir mg ) # (3.10) F ormally the susceptibility detailed in Equation 3.10 is a quantum mechanical object that appears to not ha v e a well dened classical limit ( ~ 0 ). T o de v elop a theory that is a menable to classical or semiclassical computation techniques, a distinct classical limit of this e xpression must be obtainable. This limit is more than just book k eeping; it establishes a deniti v e quantum-classical correspondence, and v alidates that a method using classical or semiclassical computational techniques can capture the true ph ysics of the optical process. In pursuit of establishing a well dened classical limit of (2)ij k the dimensionality of ~ must be reduced. T o accomplish this, the nature of the applied elds that are characteristic of an SVFS e xperiment are considered. As noted pre viously only the infrared eld is vibrationally resonant – both the sum frequenc y and visible elds f all f ar from resonance, and typic al SVFS e xperiments are also electronically nonresonant. W ith these 21

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considerations in mind, terms in Equation 3.10 that will dominate when the frequenc y of the infrared eld becomes vibrationally resonant with a mode in noncentrosymmetric media can be identied. This observ at ion f acilitates writing (2) in terms of a resonant ( (2) R E S ij k ) and nonresonant ( (2) N R ij k ) contrib ution. (2)ij k = (2) R E S ij k + (2) N R ij k (3.11) (2) R E S ij k = X mng (0)g g ~ 2 kmg ( I R mg + ir mg ) ig n jnm ( s ng + ir ng ) jg n inm ( v is + ng + ir ng ) + jnm img ( s + mg + ir mg ) inm jmg ( v is mg + ir mg ) kg n ( I R + ng + ir ng ) !# (3.12) (2) N R ij k = X mng (0)g g ~ 2 jg n img knm ( s + mg + ir mg )( v is + ng + ir ng ) + ig n jmg knm ( s ng + ir ng )( v is mg + ir mg ) # (3.13) From here on, focus will be only on (2) R E S ij k because it pro vides the most information pertaining to the resonant vibrational lineshape. Because both the signal and visible elds f all f ar from resonance, it is approximated that 1/ s 1 =! v is This approximation allo ws for the ~ dimensionality of (2) R E S ij k to be reduce by one order through the introduction of the system polarizability operator The system polarizability operator is essentially the rst order susceptibility and is dened as: ij ( ) = 1 ~ X g ;n ig n kng + ng ir ng + kg n ing + ng + ir ng # (3.14) 22

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Utilizing Equation 4.13, the resonant second order susceptibility can be written as a sum of tw o terms. (2) R E S ij k = 1 ~ X mg (0)g g pq g m kmg ( I R mg + ir mg ) kg n pq ng ( I R + ng + ir ng ) (3.15) The resonant denominators in Eq uation 3.15 can each be replaced using inte gral identities: R 1 0 dt e it ( ! o ir ) = i ! o ir and R 1 0 dt e it ( + o + ir ) = i + o + ir Because r is a phenomenological damping f actor that is naturally incorporated into the dynamics of the system, the implied limit that g amma goes to zero is also tak en. Perform ing these steps allo ws Equation 3.15 to be written as: R es ij k = i ~ Z 1 0 X g m e i! mg t e i! I R t ij g m kmg dt X ng e i! ng t e i! I R t ij ng kg n dt # (0)g g (3.16) R es ij k = i ~ Z 1 0 dt e it I R < ij ( t ) k (0 ) > Z 1 0 dt e it I R < k (0 ) ij ( t ) > (3.17) Equation 3.17 follo ws as an e xact re write of Equation 3.16, and e xpresses the SVFS resonant second order susceptibility in terms of the cross correlation of the system dipole and polarizability operators. Note, in deri ving Equation 3.17 from 3.16, the Heisenber g representation of the time dependent system polarizability operator ij ( t ) w as used, and a sum o v er states w as performed to remo v e a resolution of the identity [24, 37] Further simplication of Equation 3.17 is possible because the tw o TCF' s it is comprised of are comple x conjug ates – < k (0 ) ij ( t ) > = C R ( t ) + iC I ( t ) = ( < ij ( t ) k (0 ) > ) [37] Expressing the TCF' s in Equation 3.17 e xplicitly as the sum or dif ference of their 23

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real, C R ( t ) and imaginary C I ( t ) components equates the resonant SVFS susceptibility with an e xponential inte gral o v er the imaginary component of the correlation function. R es ij k ( I R ) = 2 ~ Z 1 0 dt e it I R C I ( t ) (3.18) Equation 3.18 presents R es ij k as an e xplicit function of the IR frequenc y because the other of f-resonant optical frequencies, v is and s ha v e been implicitly absorbed into the polarizability Equation 3.18 is a nearly e xact re write (e xact other than substituting 1/ s 1 =! v is ) of the perturbation e xpression, b ut there is still one order of ~ that must be eliminated to establish a well dened classical limit of the resonant SVFS second order susceptibility T o accomplish this, the TCF' s in Equation 3.17 will be considered in their corresponding frequenc y space representation. Note, C I is odd in frequenc y and time, and C R is e v en in frequenc y and time. [37, 62, 63] Thus, while C ( t ) is comple x, C ( ) is purely real. C ( ) = X g n Z 1 1 dt e i! t g g n ng e i! ng t = 1 Q X g n e ~ g g n ng ( ! ng ) (3.19) C ( ) = X g n Z 1 1 dt e i! t g g n ng e i! ng t = 1 Q X g n e ~ g g n ng ( + ng ) (3.20) Letting n $ g : C ( ) = 1 Q X g n e ~ n g n ng ( ! ng ) (3.21) 24

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Equation 3.21 dif fers from Equation 3.19 only in the Boltzmann f actor and if the delta function is enforced, a detailed bal ance relationship between C ( ) and C ( ) can be identied. C ( ) = e ~ C ( ) (3.22) Proceeding with some additional algebra: C ( ) = C R ( ) + C I ( ) = e ~ ( C R ( ) C I ( )) (3.23) C R ( ) e ~ 1 = 1 + e ~ C I ( ) (3.24) C I ( ) = tanh ~ 2 C R ( ) (3.25) The relationship detailed by Equation 3.25 between the real and imaginary parts of the correlation function address es tw o ph ysical constraints of the problem. First, ho w does one calculate the imaginary component of a TCF using classical MD? This relationship equates the only quantity that can be computed using classical MD, C R ( t ) and the formal object tha t describes the spectroscop y C I ( t ) Secondly a deniti v e classical limit of the susceptibility must be obtainable if the true ph ysics of the optical process can be captured using classical or semiclassical computation techniques. In the limit ~ tends to w ard zero, tanh ~ 2 ~ 2 Thus, all orders of ~ cancel, and a well dened classical limit is obtained. Equation 3.26 summarizes the nal relationship, and is the principle result of this section. R es ij k ( I R ) = 2 ~ Z 1 0 dt e it I R tanh ~ 2 C R ( t ) ! Z 1 0 dt e it I R C R ( t ) (3.26) 25

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Before concluding thi s section, and discussing the application of the theory de v eloped herein, it s hould be highlighted that the relationship detailed by Equation 3.25 is completely general. It is v alid for an y tw o-point, one-time, correlation function. Ho we v er note that because the sign of the forw ard and re v erse F ourier transfers often dif fer from one te xt to the ne xt (despite there being a formally correct quantum mechanical sign requirement), sometimes there is a m inus sign associated with this tanh detailed balance relationship – as long as all forw ard and re v erse F ourier transforms are consistent within this type of (c yclical) deri v ation, both con v entions will produce the same answer 3.2 SVFS – A pplication of Theory & Computational Methods In order to calculate the time dependent spectroscopic observ ables inherent in the usual SVFS TCF a spectroscopic model that supplements the MD force elds must be established. The foundations of our basic spectroscopic model (BSM) ha v e been presented in Chapter 2. Briey the permanent dipole, polarizability and their deri v ati v es for each species present in an MD simulation is parametrized as a functi on of molecular geometry via detailed electronic structure (ES) calculations or e xperiment. T o account for induced dipoles and polarizabilities arising from interatomic interactions, a point atomic polarizability model of the Thole-Applequist interaction model (referred to here as P AP A) type is used. [6, 18, 19, 34, 43, 44] The P AP A model' s accurac y arises from its nat ural incorporation of the intrinsic dipo le and polarizability parameters and the e xplicit incorporation of condensed phase interactions between the polarizable sites. 26

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Pre vious studies [6, 18, 19, 38, 64, 65], ha v e v alidated the ef fecti v eness of using this semiclassical computation technique (cla ssical MD + BSM) to calculate the spectroscopic signatures of comple x liquids and interf aces. In the conte xt of SVFS, man y of these pre vious studies ha v e focused on interf acial w ater and ha v e al lo wed for substantia l ne w ph ysical ins ight into this system to be realized. F or e xample, a no v el vibrational mode around 875 cm 1 present e xclusi v ely at the w ater/v apor interf ace w as disco v ered. [6, 19] Other recent e xperiments [66, 67] and theory [68] ha v e indirectly inferred the presence of this no v el surf ace species – a w ater molecule with tw o dangling h ydrogens. The reason e xperimental studies ha v e only been able to infer the e xistence of this species is purely due to technological barriers – there is a lack of intense tunable IR radiation sources in the lo wer frequenc y range. [69–71] Studies using classical MD supplemented with the BSM are capable of not only accurately capturing a syst em' s interf acial or b ulk resonant lineshape b ut the ph ysically signicant ne details within this lineshape. Specically in conte xt of SVFS, distinct subpopulations that occur at a w ater/v apor interf ace and collecti v ely contrib ute to the OH stretching spectral re gion ha v e been identied. [6, 19] This w as accomplished via e xploiting the absorpti v e and dispersi v e nature of the calculated real and imaginary components of the resonant SVF S susceptibility The frequencies of the identied subspecies compare e xceptionally well with those observ ed by tw o separate e xperimental groups. [48, 72, 73] The a v erage de viation bet ween the theoretical and e xperimental determinations w as less than 1%. [6, 19] 27

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Collecti v ely the data presented e vidences the rob ustness of the BSM, and its ability to capture the comple x man y-body interactions present in a condensed phase system. (In neat liquid w ater nonetheless!) Ho we v er in tting the point electric moments and electric moment deri v ati v es for a gi v en isolated g as phase molecule, there is not necessarily a unique solution. This is the case with w ater Se v eral solutions ha v e been found for w ater that appear to reproduce both the infrared, Raman, and SVFS homodyne detected spectra. Ho we v er v ery recent e xperimental w ork [73] w as thought to ha v e re v ealed an incorrect change of phase in our past calculation of the real and i maginary com ponents of the susceptibility for the w ater/v apor interf ace in the OH stretching re gion. All phase information outside of the OH stretching re gion matched e xperimentally obtained spectra. This pre viously went unnoticed because, due to technological barriers; only the square modulus of the SVFS signal w as able to be e xperimen tally obtained. Gi v en that our semiclassical computat ion techniques are able to reproduce such ne detail as the location of subpopulations of vibrational species, the phase error w as thought to be due to choosing the wrong s olution obtained from tting the point polarizability and polarizability deri v ati v es. Thus, this moti v ated retting the BSM for w ater based upon ES calculations of an isolated w ater molecule. All ES calculations were done using PCGAMESS at the aug-cc-pvqz basis set le v el, and electron correlation w as accounted for using MP2. The polarizability tensor w as calculated using ES, and the se v alues were used to parametrize our w ater model. [6, 19, 74] Specically an algorit hm w as implemented to solv e for the point polarizabilities of Oxy28

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ES x y z O 0. 0. -0.065565653769 H1 0.7569498491 0. 0.5202848504 H2 -0.7569498491 0. 0.5202848504 rOH 0.957180 T able 3.1: The equilibrium conguration of an isolated w ater molecule used in ES calculations. Units are in Angstroms. gen and Hydrogen such that when the electric moments were allo wed to interact via the BSM, the polarizability tensor obtained from the ES calculation w as reproduced. T ables 3.1 and 3.2 detail the w ater conguration used in the ES and BSM calculations respecti v ely Note, the congurations betwe en ES and BSM v ary This is because the BSM uses the equilibrium conguration of the MD model, whereas the ES conguration is obtained from e xperiment. It w as found that se v eral solutions were capable of reproducing the polarizability tensor These solutions were then further tested by assessing their capability to reproduce the polarizability deri v ati v e matrix. Constructing the polarizability deri v ati v e matrix using our spectroscopic model in v olv ed making 9 independent calculations. In each calculation, a single coordinate w as displaced by 0.00001 A (in the x, y or z directions of the Oxygen and 2 Hydrogens) from the equilibrium conguration of w ater The set of point polarizabilit ies (Oxygen=1.307 A 3 =e Hydrogen=0.157 A 3 =e ) that naturally best captured the deri v ati v e matrix as dened by the ES calculations w as 29

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BSM x y z O. 0. 0. 0. H1 0.8179551669 0. 0.5752961881 H2 -0.8179551669 0. 0.5752961881 rOH 1.0 T able 3.2: The equ ilibrium conguration of an isolated w at er molecule used in the basic spectroscopic model (BSM) calculations. Units are in Angstroms. ES x y z BSM x y z x 1.512355 0. 0. x 1.562470 0. 0. y 0. 1.441227 0. y 0. 1.31758 0. z 0. 0. 1.469676 z 0. 0. 1.424445 h i 1.47442 h i 1.434832 T able 3.3: The polarizabilit y tensor ( A 3 / e ) of an isolated w ater molecule calculated via ES and BSM. 30

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ij xx yy zz xz=zx xy=yx yz=zx Ox:ES – – – 1.080 – – Ox:BSM – – – 1.080 – – %Error – – – 0. – – Oz:ES -1.479 -0.463 -1.282 – – – Oz:BSM -1.508 -0.475 -1.254 – – – %Error 1.961 2.592 -2.184 – – – H1x:ES 1.281 0.401 0.672 – – 0.543 H1x:BSM 1.345 0.339 0.635 – – 0.540 %Error 5.00 15.461 5.82 – – 0.552 H1z:ES 0.691 0.230 0.575 – – 0.420 H1z:BSM .754 0.238 0.627 – – 0.464 %Error 9.117 3.478 -8.293 – – 10.476 H2x:ES -1.388 -0.405 -0.803 – – -0.543 H2x:BSM -1.345 -0.339 -0.635 – – -0.540 %Error 3.098 16.296 20.922 – – -0.552 H2z:ES 0.691 0.230 0.575 – – 0.420 H2z:BSM 0.754 0.238 0.627 – – 0.464 %Error 9.117 3.478 9.043 – – 10.476 T able 3.4: Polarizability deri v ati v e matrix. ES and BSM are compared for the 9 coordinate displacements. Only deri v ati v es with an order of magnitude 0.1 A 2 =e are presented. 31

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selected to be furthe r rened. T able 3.3 details the polarizability matrix the selected set of point polarizabilities produced, and compares them with ES polarizability v alue s. T o the selected set of point polarizabilities, point p olarizability deri v ati v es were t as a function of displacement from equil ibrium bond length: = o + 0 r The parameters determined for the point polariza bility deri v ati v es were Oxygen=.353 A 2 =e and Hydrogen=.885 A 2 =e T able 3.4 gi v es the deri v ati v e polarizability tensor obtained from both the BSM and ES ca lculations along with the percent error of the BSM polarizability deri v ati v e matrix. Note, the polarizability deri v ati v e matrix obtained via ab initio calculations used the same coordinate displacement method detailed pre viously 32

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Figure 3.1: Experimental SVFS results for the w ater/v apor interf ace. T op P anel: Resonant homodyne signal. Middle P anel: Real component of the resonant signal. Bottom P anel: Imaginary contrib ution to the resonant signal. In all cases the x-axis is in w a v enumbers, and the y-axis is in arbitrary units. 33

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Figure 3.2: Calculated spectra of the w ater/v apor interf ace using the reparametrized spectroscopic model. T op P anel: Comparison of the real (red) and imaginary (green) components of the SVFS resonant signal. The lines represent the e xperimentally obtained locations of subpopulations: 3195 cm 1 (blue), 3325 cm 1 (purple), 3400 cm 1 (yello w), 3500 cm 1 (tan), 3694 cm 1 (black). In the calculated spectra, a subpopulation is identied by ha ving one of the lineshapes e xpress an absorp ti v e characteristic and the other possess a disp ersi v e characteristic in the same spectral range. Bottom P anel: Calculated resonant homodyne signal. 34

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Figure 3.3: Calculated spectra of the w ater/v apor interf ace using the old spectroscopic model. T op P anel: Comparison of the real (red) and imaginary (green) components of the SVFS resonant signal. The lines represent the e xperimentally obtained l ocations of subpopulations: 3195 cm 1 (blue), 3325 cm 1 (purple), 3400 cm 1 (yello w), 3500 cm 1 (tan), 3694 cm 1 (black). In the calculated spectra, a subpopul ation is identied by ha ving one of the lineshapes e xpress an absorpti v e characteristic and the other possess a dispersi v e characteristic in the same spectral range. Bottom P anel: Calculated resonant homodyne signal. 35

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Figures 3.1 – 3.3 sho w the lineshapes obtained from the ne w model, the old model, and the latest e xperimental data collected by the Shen group. [73] The real and imaginary spectra calculated via the ne w model appear to be in much better agreement with the e xperimentally obtained real and imaginary lineshapes. Ho we v er the intensity of the calculated homodyne (square modulus) signal is not correct for the ne w mo del, and is better captured by the pre vious model. This discrepanc y can be e xplained. In the old w ater model, the polarizability deri v ati v es where chosen based upon their ability to produce the most intense fr ee OH stretching peak when calculating the homodyne SVFS w ater/v apor interf ace spectrum. (Due to technol ogical limitations, this w as pre viously the only quantity that could be measured e xperimentally .) The ne w polarizability deri v ati v e parameters were chosen because of their abil ity to accurately reproduce the g as phase w ater polarizability deri v ati v e tensor Because the intensity of a lineshape is inherently tied to quantum mechanical phenomenon, the ne w model, while semiclassical in nature and also emplo ying a quantum correc tion technique, must not be capturing the true population of states. The old model w as unph ysically adjusted to do so in the OH stretching re gion, b ut making this unph ysical adjustment in the old model af fected other portions of the lineshape also. Namely this is wh y the peak around 3400 cm 1 lacks intensity in the spectrum produced via implementation of the old model. It is important to note, that while not capturing all the quantum mechanical ph ysics of the system, our semiclassical method is quite rob ust as e videnced by its ability to reproduce the correct man y-body interactions in liquid w ater that result in a sizable shift in fundamental res36

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onant vibrational frequenc y in the condensed phase as compared to the g as phase. (The spectroscopic model w as t to an isolated w ater molecule.) Secondly one must be mindful when comparing the intensity of calculated and e xperim entally obtained lineshapes. This is because the intensity of the e xperimentally obtained SVFS homodyne spectra is strongly inuenced (roughly proportional to the square of the frequenc y a resonant peak appears at) by necessary e xperimental parameters, and there is no general con v ention for remo ving these ef fects from the detected signal. [3, 4, 6, 75] When decomposed into the real and imaginary spectra, the error in intensity reduces to roughly order Due to a distinct combination of the tw o phenomenon discussed, this is wh y the real and imaginary spectra calculated with the ne w model are in good agreement with e xperimental data, and the square modulus signal possesses the correct lineshape b ut incorrect relati v e intensity Ne glecting the intens ity dif ferences, Figures 3.1 – 3.3 are all in agreement with respect to subpopulation location and phase. What about the apparent phase error that moti v ated the complete reparametrization of a ne w spectroscopic model for w ater? It w as found that the apparent phase error w a s a result of a restraining potential used in the MD calculations to pre v ent a w ater molecule starting in one half of the simulation box from dif fusing to the other side' s interf ace – which w ould result in signal cancellation. [6, 19] Thus, if the restraining potential is not applied, man y TCF' s c omputed for a duration of time less than the time it w ould tak e for a w ater molecule to dif fuse half the length of the simulation box is necessary to obtain an a v eraged TCF that is easily F ourier transformed. In past studies [6, 38] that did not use a restraining potential, this w as the general methodology 37

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for calculating SVFS spectra. Ho we v er whi le t he a v erage of the man y short time TCF' s obtained were F ourier transformable, the signal w as noisy to the e xtent that the location of subpopulations could not be identied. The reason wh y using a lateral restraining potential centered in the middle of the box creates a signal with t he wrong phase in only an isolated re gion of the spectrum is because its presence creates an articial interf ace that mimics an en vironment that is similar in nature to the bottom of the rst condensed phase w ater layer at the w ater/v apor interf ace. The interf ace created by the restraining potential is the opposite en vironment (directionally) of the w ater/v apor interf ace, and thus there is competiti v e (deconstructi v e) interference in the calculated signal from these tw o en vironments. Further it has been sho wn [76] that the bottom of the rst condensed phase w ater laye r primarily contrib utes to the 3400 cm 1 re gion, while the top layer accounts for the free OH species. The ph ysical constraints of ha ving a restraining potential do not allo w for a free OH species to be created. This wh y the phase of the free OH w as not af fected. The ne w phase correct, spectra whe re obtained by ne glecting the molecules in the middle third re gion of the simulation box when calculating the SVFS TCF This e xclusion creates a boundary in the box, b ut not an interf ace. Ef fecti v ely this boundary acts as a poor man' s implicit solv ation. 38

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3.3 Concluding Remarks on New W ater Model & Futur e Impr o v ements In analyzing the polarizability deri v ati v e tensor the a v erage percent error between the ES and BSM is 6.882. The tted BSM for w ater reproduces the general polarizability deri v ati v e trends, b ut the percent error of se v eral of these elements is abo v e 10%. This leads to some concern. Ho we v er gi v en that the IR, Raman, and phase correct SVFS spectra were reproduced from these parameters, further renement w as not pursued. In the future, it may be desired t o ret the polarizability v alues for the BSM of w ater as a function of displacement from equilibrium bond length and displacement from the equilibrium bond angle ( ): ( r ; ) = o + 0 r r + 0 Another possibility w ould be to not approximate the polarizability and/or polarizability deri v ati v es to be points, b ut rather tensors as the y are formally gi v en. This later method could the n also be parametrized as a function of displacement from equilibrium bond length and/or bond angle. 39

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Chapter 4 Quadrupole Induced Bulk SVFS As described in Chapter 3, under the usual dipole a pproximation, second order (threew a v e mixing) spectroscopies v anish in isotropic media due to the in v ersion symmetry of such systems. [2, 3, 6] Interf aces serv e to break this symmetry and produce a second order polarization signal. Be yond the dipole approximation, the b ulk of a s ystem can contrib ute coherently to se cond order optical measurements through quadrupole (and higher order) ef fects. While quadrupole contrib utions can be se v eral orders of magnitude smaller than dipole ef fects, [30] the relati v e number of absorbers in the b ulk vs. interf acial re gions is lar ge – making the collecti v e quadrupole contrib ution signicant in some systems. Note, the quadrupole contrib utions from the b ulk originate in a re gion that is roughly the w a v elength of light used while the interf acial contrib ution is li mited to a fe w molecular layers. The dynamics between these tw o re gions can be signicantly dif ferent. [6, 19] There ha v e been a multitude of careful and interesting interf acial studies of liquid systems – both e xperimentally [12, 53, 55, 58, 60, 69, 72, 77–81] and theoretically [6, 19, 38, 50, 68, 82–84] – using SVFS. These studies ha v e typi cally assumed the dipole approximation 40

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to be adequate in either interpreting or calculating the SVFS signal. Quadrupole contrib utions ha v e usually a priori been assumed to be ne gligible because the y are minimized by taking e xperimental optical measurements in the reected (as opposed to transmission) geometry (Figure 4). [13] Shen et. al ha v e sho wn there are no established general ph ysical criteria that determine when quadrupole contrib utions can be ne glected, and that the y need to be in v estig ated on a case by case basis. [53, 85, 86] Further while e xper imental determinations (assessing the dif ferences in the signal in the transmission and reection geometry) can determine the relati v e importance of the b ulk signal, it is not possible to completely separate the b ulk and surf ace contrib utions. [86] Also, the importance of b ulk contrib utions is highly dependent on the particular polarization condition that is probed. [85, 86] Thus, in the goal of accurately interpreting three-w a v e mixing spectra, it is to the be net of both the e xperimental and theoretical communities to ha v e a general molecularly detailed technique by which quadrupole contrib utions to SVFS spectra can be quantied. Therefore, a molecularly detailed TCF approach for calculating b ulk quadrupole contrib utions to the SVFS spectra that is generally v alid has been de v eloped, and is presented in detail belo w This chapter concludes with a practical, computationally tractable, implementation model of the theory including permanent and induced quadrupole ef fects. 41

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Figure 4.1: Coplanar geometry of the incident, reected and transmitted beams. 1 ( 2 ) is the angle of incidence with respect to the z -axis of the visible (IR) eld. S F G ( D F G ) is the angle the generated SFG (DFG) signal is radiated at k 1 ( k 2 ) is the w a v e v ector of the visible (IR) eld. k s r ( k s T ) is the w a v e v ector of the reected (transmitted) eld, and k s = k 1 + k 2 All incident elds are assumed to lie in the same xz plane which is normal to the surf ace. 42

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4.1 General Theor etical De v elopment of Quadrupole Contrib utions The N th order polarization signal a gi v en spectroscop y measures is proportional to the N th order susceptibility of a system. [2, 3, 6] Therefore, de v eloping a TCF theory of quadrupole contrib utions to the SVFS spectra requires starting with the general second order susceptibility e xpression. This process de viates from the theoretical de v elopment detailed in Chapter 3 in that it must incorporates both dipole and quadrupole ( q ) contrib utions in the perturbed Hamiltonian, H (1) This implies that the total second order polarization is gi v en by: P (2) ( r ; t ) = M h (2) i + M 5 h q (2) i (4.1) T o obtain a general e xpression for the susceptibility that includes both dipole and quadrupole contrib utions, the orders of the density matrix must be resolv ed. Rederi ving (1) (2) and the no w multiple (1) components using H (1) = E ( t ) q 5 E ( t ) the total second order polarization in terms of tensor components, including both dipole and quadrupole contrib utions, can be e xpressed as: P (2) i = P (2) ;D i + 5 j P (2) ;Q ij (4.2) P (2) ;D i = (2) D ij k E j ( q ) E k ( p ) + (2) D q 1 ij k l @ E j ( q ) @ r k E l ( p ) (4.3) + (2) D q 2 ij k l E j ( q ) @ E k ( p ) @ r l + (2) D q 3 ij k l m @ E j ( q ) @ r k @ E l ( p ) @ r m P (2) ;Q ij = (2) Q ij k l E k ( q ) E l ( p ) + (2) Q q 1 ij k l m E m ( p ) @ E k ( q ) @ r l (4.4) + (2) Q q 2 ij k l m E k ( q ) @ E l ( p ) @ r m + (2) Q q 3 ij k l mn @ E k ( q ) @ r l @ E m ( p ) @ r n 43

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In Equations 4.2 – 4.4 the Einstein summation notation is implied. Here, P (2) i is the total second order polarization. P (2) ;D i and P (2) ;Q ij are the dipole and quadrupole moment contrib utions to the tota l second order polarization respecti v ely P (2) ;D i contains terms that collecti v ely contrib ute to P (2) i linearly and P (2) ;Q ij contains terms that collecti v ely contrib ute to P (2) i through P (2) ;Q ij s gradient. Comparison of the second order dipolar polarization e xpression deri v ed in Chapter 3 with the components of Equation 4.2 re v eals that only the rst term in Equation 4.3 is obtained when quadrupole contrib utions are ne glected. Hence, all other terms in Equation 4.2 are inherently quadrupole in origin. Note, the nal term in Equation 4.3 and the nal three terms in Equation 4.4 ha v e generally been ne glected in the literature when quadrupole contrib utions ha v e been discussed because the y are higher order contrib utions in the sense that t he y in v olv e multiple gradients, [87, 88] b ut can be important, especially when considering systems containing ph ysically lar ge components – such as, metallic systems, suspended nanoparticles, or colloids. [14, 15] The follo wing general e xpressions for the susceptibilities in Equations 4.3 and 4.4 are gi v en in a form that suppresses the required intrinsic permutation symmetry for bre vity; to deri v e the TCF e xpressions that follo w the full susceptibility tensors, including intrinsic permutation symmetry must be considered. K eeping with the notation established in pre vious chapters, iab ( q i ab ) is a dipole (quadrupole) matrix element between states a and b with a polarization component of i ab a b and r ab = r ba r is a damping f actor that controls the line width, and is naturally incorporated into the system dynamics. 44

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(2) D ij k = M ~ 2 X g r v (0)v v ig v kv r jr g ( s + g v + ir g v )( p + r v + ir r v ) + iv g kr v jg r ( s g v + ir g v )( p r v + ir r v ) ir g kv r jg v ( s g r + ir g r )( p + r v + ir r v ) ir g kg v jv r ( s g r + ir g r )( p g v + ir g v ) (4.5) (2) D q 1 ij k l = M ~ 2 X g r v (0)v v ig v lv r q j k r g ( s + g v + ir g v )( p + r v + ir r v ) + iv g lr v q j k g r ( s g v + ir g v )( p r v + ir r v ) ir g lv r q j k g v ( s g r + ir g r )( p + r v + ir r v ) ir g lg v q j k v r ( s g r + ir g r )( p g v + ir g v ) (4.6) (2) D q 2 ij k l = M ~ 2 X g r v (0)v v ig v q k l v r jr g ( s + g v + ir g v )( p + r v + ir r v ) + iv g q k l r v jg r ( s g v + ir g v )( p r v + ir r v ) 45

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ir g q k l v r jg v ( s g r + ir g r )( p + r v + ir r v ) ir g q k l g v jv r ( s g r + ir g r )( p g v + ir g v ) (4.7) (2) D q 3 ij k l m = M ~ 2 X g r v (0)v v ig v q l m v r q j k r g ( s + g v + ir g v )( p + r v + ir r v ) + iv g q l m r v q j k g r ( s g v + ir g v )( p r v + ir r v ) ir g q l m v r q j k g v ( s g r + ir g r )( p + r v + ir r v ) ir g q l m g v q j k v r ( s g r + ir g r )( p g v + ir g v ) (4.8) (2) Q ij k l = M ~ 2 X g r v (0)v v q ij g v lv r kr g ( s + g v + ir g v )( p + r v + ir r v ) + q ij v g lr v kg r ( s g v + ir g v )( p r v + ir r v ) q ij r g lv r kg v ( s g r + ir g r )( p + r v + ir r v ) q ij r g lg v kv r ( s g r + ir g r )( p g v + ir g v ) (4.9) (2) Q q 1 ij k l m = M ~ 2 X g r v (0)v v q ij g v mv r q k l r g ( s + g v + ir g v )( p + r v + ir r v ) 46

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+ q ij v g mr v q k l g r ( s g v + ir g v )( p r v + ir r v ) q ij r g mv r q k l g v ( s g r + ir g r )( p + r v + ir r v ) q ij r g mg v q k l v r ( s g r + ir g r )( p g v + ir g v ) (4.10) (2) Q q 2 ij k l m = M ~ 2 X g r v (0)v v q ij g v q l m v r kr g ( s + g v + ir g v )( p + r v + ir r v ) + q ij v g q l m r v kg r ( s g v + ir g v )( p r v + ir r v ) q ij r g q l m v r kg v ( s g r + ir g r )( p + r v + ir r v ) q ij r g q l m g v kv r ( s g r + ir g r )( p g v + ir g v ) (4.11) (2) Q q 3 ij k l mn = M ~ 2 X g r v (0)v v q ij g v q mn v r q k l r g ( s + g v + ir g v )( p + r v + ir r v ) + q ij v g q mn r v q k l g r ( s g v + ir g v )( p r v + ir r v ) q ij r g q mn v r q k l g v ( s g r + ir g r )( p + r v + ir r v ) q ij r g q mn g v q k l v r ( s g r + ir g r )( p g v + ir g v ) (4.12) 47

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4.2 TCF Expr essions f or SVFS Quadrupolar Susceptibilities The general second order susceptibil ities presented abo v e, including intrinsically permutated terms, represent a starting point for deri ving unique quantum me chanical TCF' s that describe the response of a particular type of spectroscop y Henceforth, focus will be e xclusi v ely on the resonant portion of the v arious second order susceptibilities. This is because these portions pro vide the dominant, and most informati v e, contrib ution to the resonant SVFS spectral lineshape. In this deri v ation, we are guided by the f act that all time domain response functions must be purely real. [5, 89] T o reca st the second order susceptibility tensors in terms of correlation functions describing SVFS, the dipole ( ), dipole-quadrupole ( ; ~ ), and quadrupole ( ) polarizabilities, as dened by the rst order solution to the density matrix (deri v ation not sho wn), will be used. These are dened as: ab ( ) = 1 ~ X v n v v av n bnv nv ir nv + bv n anv nv + + ir nv (4.13) ~ abc ( ) = 1 ~ X v n v v av n q bc nv nv ir nv + anv q bc v n nv + + ir nv (4.14) abc ( ) = 1 ~ X v n v v q ab v n cnv nv ir nv + av n q ab nv nv + + ir nv (4.15) abef ( ) = 1 ~ X v n v v q ab v n q ef nv nv ir nv + q ab nv q ef v n nv + + ir nv (4.16) First, the resonant components of (2) D ij k (2) D q 3 ij k l m (2) Q ij k l and (2) Q q 3 ij k l mn are considered. In pursuit of de v eloping computationally amenable TCF e xpressions, the follo wing assumptions are made: (1) r r v + r g v r g r (2) 1/ s 1/ p and (3) (where applicable) eld 48

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deri v ati v es terms are symmetric in the sense that @ E a @ r b = @ E b @ r a Note for SVFS, the rst approximation amounts to equating the frequenc y of the sum and visible elds, and both the rst and second approximations are required to deri v e the well-kno wn [6, 50] SVFS TCF that describes the system response in the dipole approximation. Letting (2) ;R E S denote only the resonant portion of the susceptibility: (2) D ;R E S ij k = 1 ~ X v g r (0)v v jr v ik v r ( s ) ( q g v + ir g v ) + ik r v ( s ) jv r q + g v + ir g v (4.17) = i ~ Z 1 0 dt e ( i! q t ) < ik ( t ) j (0 ) > i ~ Z 1 0 dt e ( i! q t ) < j (0 ) ik ( t ) > (4.18) In deri ving Equation 4.18 from Equation 4.17, the inte gral identity i R 1 0 dt e it ( a + ib ) = 1 = ( a + ib ) is used to r eplace the denominators in Equation 4.17, the denition of the Heisenber g representation of a time dependent operator is applied to obtain ( t ) and the necessary sums o v er states are performed. TCF e xpressions for (2) D q 3 ij k l m (2) Q ij k l and (2) Q q 3 ij k l mn are deri v ed in an analogous f ashion, i.e making the same approximations necessary to de v elop Equation 4.18. (2) D q 3 ;R E S ij k l m = i ~ Z 1 0 dt e ( i! q t ) h < ~ il m ( t ) q j k (0 ) > < q j k (0 ) ~ il m ( t ) > i (4.19) (2) Q;R E S ij k l = i ~ Z 1 0 dt e ( i! q t ) < ij k ( t ) l (0 ) > < l (0 ) ij k ( t ) > (4.20) (2) Q q 3 ;R E S ij k l mn = i ~ Z 1 0 dt e ( i! q t ) < ij mn ( t ) q k l (0 ) > < k l (0 ) ij mn ( t ) > (4.21) Additional simplication of Equations 4.18-4.21 is possible by writing the comple x correlation functions, general ly denoted by C ( t ) describing the resonant susceptibilities 49

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in terms of their real, C R ( t ) and imaginary C I ( t ) parts; C ( t ) = C R ( t ) + iC I ( t ) When e xpanded out in such a manner the resonant frequenc y dependent susceptibilities can be written as a half-sided transform o v er their time dependent imaginary component. [6, 52] This satises the necessary requirement that a time dependent response function is purely real. [89, 90] As w as discussed in Chapter 3, the imaginary component of a correlation function cannot be directly calculated via classical computation techniques – it is only the classical limit of the real part of the comple x correlation function that can be calculated directly Ho we v er as sho wn in Chapter 3, all tw o-point, one-time, TCF' s ha v e an analytical detailed balance relationship between their real and imaginary parts in the frequenc y domain gi v en by: C R ( ) = cotanh ( ~ = 2) C I ( ) Substitution of this relationship into the v arious resonant susceptibility e xpressions establishes a deniti v e quantum-classical cor respondence, and pro vides a direct route for calculating the microscopic susceptibilities and spectra for systems of interest. (2) D q 1 ij k l (2) D q 2 ij k l (2) Q q 1 ij k l m and (2) Q q 2 ij k l m can also each be written in terms of a half sided transform of the dif ference of tw o TCF' s that are comple x conjug ates. The re write of these four terms requires a slightly dif ferent, b ut relati v ely similar set of approximations; s g r s g v which is reasonable under typical thermal e xperimental conditions, and eld deri v ati v es terms are assumed to be symmetric in the sense that @ E a @ r b = @ E b @ r a Note, in addition to the abo v e method, the resonant portion of (2) D ij k (2) D q 3 ij k l m (2) Q ij k l and (2) Q q 3 ij k l mn can also be written in terms of TCF' s using this set of approximations. In this case, the 50

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resulting e xpressions for their resonant susceptibilities are still described by Equations 4.18 – 4.21. The resonant susceptibility for (2) D q 1 ij k l (2) D q 2 ij k l (2) Q q 1 ij k l m and (2) Q q 2 ij k l m are gi v en by: (2) D q 1 ;R E S ij k l = i ~ Z 1 0 dt e iw q t n < ~ ik l ( t ) j (0 ) > < j (0 ) ~ ik l ( t ) > o (4.22) (2) D q 2 ;R E S ij k l = i ~ Z 1 0 dt e iw q t < il ( t ) q j k (0 ) > < q j k (0 ) il ( t ) > (4.23) (2) Q q 1 ;R E S ij k l m = i ~ Z 1 0 dt e iw q t < ij l m ( t ) k (0 ) > < k (0 ) ij l m ( t ) > (4.24) (2) Q q 2 ;R E S ij k l m = i ~ Z 1 0 dt e iw q t < ij m ( t ) q k l (0 ) > < q k l (0 ) ij m ( t ) > (4.25) Equations 4.19 – 4.25 are TCF f ormulas that are capable of describing the quadrupole contrib utions to the resonant spectral lineshape at all frequencies – a v oiding th e rotating w a v e approximation which has been pre viously used [87] to describe only high frequenc y quadrupole SVFS contrib utions. Ne xt, a no v el microscopic polarizability model is presented to permi t the calculation of the abo v e TCF' s that is compatible with MD simulations.4.3 Calculation via a Char ge-Interaction Model The BSM, which includes calculation of the total system dipole and polarizability has been pre viously outlined in Chapter 2. The tting of and application to an interf acial liquid system of the BSM w as also presented in Chapter 3. The BSM must be further 51

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e xtended to calculate the total system quadrupole, dipole-quadrupole, and quadrupole polarizabilities. In addition to the total dipole and polarizability these quantities are also necessary for calculation of quadrupole-origin SVFS TCF' s. All three ne w quantities can be determined by a no v el generalization of the BSM outlined in Chapters 2 and 3. The remainder of this section will systematically detail the equations that need to be computationally implemented within a spectroscopic model to obtain the requisite additional electric moment and electric moment polarizabilities. Note, the results presented in this section are all in terms of Cartesian tensor components (superscript Greek ind ices) for a single atom (Roman indices). The v arious molecular polarizabilities are a sum of all their atomic polarizabilities, repeated Greek indices are to be summed o v er subscripts of o denote permanent /intrinsic moments, and T ::: ij is the multipole interaction tensor gi v en by: 5 5 :: 5 (1 = r ij ) where r ij is the v ector between atom i and atom j [46, 91] E i ( E i ) is a eld (eld gradient) including both local and e xternal eld contrib utions whereas E i;o ( E i;o ) denotes an e xternal eld (eld gradient). W ithin an e xtended BSM (also referred to as a P AP A model), the i nduced dipole is gi v en by Equation 4.26. Be yond the dipole approximation, the indi vidual induced dipole moments also ha v e a quadrupole contrib ution. This contrib ution is represented by the third term of Equation 4.26. [92] In the common dipole approximation e xpression, only the rst tw o terms on the right hand side are obtained. i = E i = i;o E o + i;o X j T ij j i;o 3 X j T r ij q r j (4.26) 52

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From Equation 4.26, a super -matrix detailing the ef fecti v e polarizability ij between e v ery atom pair in the system can be solv ed for [46] The induced quadrupole is gi v en by: q i = r i E r i + r i E r i (4.27) Notice, the induced quadrupole includes contrib utions from both the dipole-quadrupole, and pure quadrupole, polarizabilities. The induced dipole-quadrupole and pure quadrupole moments ha v e been demonstrated to be important in, e .g ice. [93] The induced dipole-quadrupole contrib ution to the quadrupole and dipole-quadrupole polarizability can be calc ulated in terms of Equations 4.28 and 4.29. [46, 94] Alternati v ely the numerical deri v ati v e of the total quadrupole with respect to the eld can be tak en to obtain t he dipole-quadrupole polarizability The necessa ry modication of Equation 4.29 to obtain ~ r (see Equations 4.15 and 4.14) is straightforw ard. r i E r i = r i E r i;o r i X j T r ij j r i 3 X j T r ij q j (4.28) r ij = 3 2 r i r i + 3 2 r i r i 1 2 r i r i (4.29) The induced quadrupole due to the quadrupole polarizability and quadrupole polar izability are gi v en by Equations 4.30 and 4.31 respecti v ely [94, 95] Alternati v ely the numerical deri v ati v e of the total quadrupole with respect to the eld gradient can be tak en to obtain the pure quadrupole polarizability r i E r i = r i E r i;o + r i X j T r ij j r i 3 X j T r ij q j (4.30) 53

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2 r i = r i r i + r i r i 2 3 r i r i + r r i i + r i r i 2 3 r i i r 3 2 r i r r i i 3 2 r i r i r i + r i r i r i 3 2 r i r r i i 3 2 r i r i r i + r i r i r i + r i r r i i + r i r i r i 2 3 r i r i r i (4.31) In Equation 4.30, the multipole interaction tensors T r ij and T r ij are third and fourth tensors respecti v ely for e v ery atom interaction pair ij The se can be computationally e xpensi v e to calculat e. Ho we v er the computational e xpense can be signicantly reduced by: (1) e xploit ing the non-uniqueness of man y of the elements present in the multipole interaction tens or for a gi v en rank, and (2) using a recursi v e relationship to generate the N rank interaction tensor from the N 1 rank tensor [96] Additionally symmetries are present in the dipole-quadrupole polarizability: r (pure-quadrupole pola rizability r ) is symm etric in and ( and r and ). [94] Despite man y simplications, the computational implementation of thi s e xtended BSM is a sizable project. Thus, it will be the subject of future w ork and continued ongoing renements to the Space Research group spectroscopic model. 54

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Chapter 5 Static Field Induced Third Order SVFS Liquid interf aces are ab undant in chemistry and the en vironment, and it is common to ha v e char ged species or char ged solids surf aces present at these interf aces. Char ged species can be, e .g surf actants or an y other amphiphile, and a ubiquitous e xample of a char ged solid/liquid interf ace is the silica/w ater interf ace. [97–99] Silicates are common in both the soil and atmospheric dust where man y important chemical processes occur [100, 101] Interf aces of this nature (silica/w ater) play a critical role in binding pollutants and biological molecules, such as he xa v alent chromium, the agricultural antibiotic Morantel, and or g anic phosphate compounds. [79, 99, 102–104] The char ge associ ated wi th the silica/w ater interf ace is primarily due to the silanol groups, SiOH, w hich terminate silica. These groups ionize in w ater – especially as pH is increased, [102, 105, 106] and e v en the undissociated silica surf ace produces a relati v ely lar ge static eld due to lar ge char ge separation between the atoms (silicon, oxygen, and h ydrogen). [102, 104] Gi v en the pre v alent nature of char ged interf aces, and the increasing use of second or der optical techniques to interrog ate interf aces, a question naturally arises: ho w does the 55

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static eld associated with a char ged surf ace modify what an SVFS e xperiment is fundamentally probing? While a mathematical description of SVFS at a char ged interf ace will be address ed in the follo wing sections, the answer to this ques tion is easy to conceptualize. Specically the presence of a static eld associated with a char ged interf ace will both produce a more intense second order signal (due to ordering at the interf ace), and a distinct third order signal (due to contrib utions from re gions into which the static eld penetrates). The ca v eat of SVFS at char ged interf aces is that these tw o distinct contrib utions are unable to be independently resolv ed e xperimentally b ut indee d ha v e dif ferent ph ysical origins. [6, 19, 34, 96] Further there is suggesti v e e vidence in the literature of the importance/relati v e magnitude of this third order contrib ution, b ut no pre vious molecularly detailed approach e xisted to separately calculate the second and third order contrib utions. The rst molecularly detailed TCF approach that allo ws for the second and third order contrib utions to SVFS to be indi vidually determined is presented in t he follo wing section. Practical implementation procedures of the deri v ed TCF' s that descri be these separate phenomenon is subsequently discussed. The Chapter concludes with results of the discussed model applied to a w ater molecule and a w ater dimer system. 5.1 Effecti v e P olarization Due to A Static Field in Isotr opic Media In the limit of monochromatic elds, the ef fecti v e observ ed polarization, P ef f for a general second order optical e xperiment at a char ged interf ace (typical referred to as Electric 56

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Field Enhanced SHG/SFG/DFG) is gi v en by: [105–110] P ef f = P (2) + P (3) = (2) j E 1 E 2 + (3) j E 1 E 2 E static (5.1) Here, E static repres ents the static eld, and is completely general at this point; meaning it can be e xternally applied or an intrinsic characteristic of chemical medium. [105–107, 111] (It should be noted that spectroscopic t echniques emplo ying e xternal static electric elds ha v e been used in int erf acial studies of solids since the 1960' s, [23, 112] b ut it has not been until more recently that analysis of liquids at intrinsically char ged inter f aces ha v e become more common. [105, 108, 110, 111, 113–115]) In Equation 5.1, both the second order and third order polarizations contrib ute coherently to P ef f and directly probe the second and third order susceptibilities respecti v ely Thus, Equation 5.1 implies the measured polarizat ion signal, in addition to the normal second order signal, contains tw o other signicant contrib utions. (1) The presence of three elds (tw o incident + static) gi v es rise to a third order nonlinear polarization which is not strictly interf ace specic. (2) The symmetry is brok en by the presence of E static and, thus, i t further e xtends the anisotropic interf acial re gion into normally centrosymmetric re gions of the b ulk. [107, 110] (1) directly probes the t hird order susceptibility (2) results in an intensication of the second order susceptibility [105, 106] Hence, these additional contrib utions to the observ ed polarization are sensiti v e to the e xtended interf acial re gion resulting from the static eld. [107] The adv antage of electric eld enhanced three-w a v e mixing in v estig ations lies is th eir ability t o deduce the electrostatic potent ial created by the char ged species near the interf ace, and monitor ho w their electrostatic potential changes the na57

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ture of the interf ace; [105, 106] this capability critically relies on separating the second and third order polarization contrib utions. Note, P ef f constitutes a combined measurement of second and third order processes due to the w a v e v ector of the static eld being zero. A eld' s w a v e v ector is proportional to its frequenc y and the e xperimentally detected polarization signal is determined by the sum of the perturbing elds' w a v e v ectors. [2, 4, 6] Thus, the tw o perturbing applied SHG/SFG/DFG elds ha v e non-zero w a v e v ectors, k 1 and k 2 while the static eld has a w a v e v ector of zero. This implies the P (2) and P (3) signals are generated in the direction: k sig n a l = k 1 + k 2 and k sig n a l = k 1 + k 2 + 0 respecti v ely 5.2 Micr oscopic (3) Expr ession to Account f or a Static Field Frequenc y domain perturbati v e e xpressions describing the N th order susceptibility can be found in the literature, [2–4] and can be used as the starting point in de v eloping a TCF theory [6, 50, 87] Ho we v er when an y of the elds are of zero frequenc y the e xpressions describing the N th order susceptibility appear to contain terms that, under s pecic conditions, can become secular di v er gences. (These apparent di v er gences can also occur when an y set of the applied elds frequenc y' s sum to zero.) The case of secular di v er gence is distinct from analyzing perturbat i v e e xpressions for di v er gent terms to assess the dominant/resonant contrib utions to the N th order susceptibility Specically when eld frequencies sum to zero, no resonance condition is met, and it is termed a secular di v er gence. 58

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Consider for e xamp le the typical form of denominators in frequenc y dependent susceptibility e xpressions: 1 = ( r g i ir r g ) While both the secular and resonant terms are di v er gent, a resonance occurs when the e xperimental eld(s), i is of the same frequenc y as a transition, r g – both of which are indi vidually nonzero. A secul ar di v er gence appears in this e xample when i is of zero frequenc y and the transition between states is also of zero frequenc y – i.e., r g = r g r g = 0 As sho wn by Y uratich [116] and W ard [117] et. al using the Method of A v erages, the apparent secular nature of ( N ) when an y subset of the applied elds sum to zero v anishes. In our analysis of the perturbati v e e xpression for SVFS (3) including al l forty-eight terms (e xpression not sho wn), in the presence of a static eld, we ha v e also found all secularly di v er gent terms that w ould appear to contrib ute to the resonant susceptibility e xactly cancel out one another While this result w as e xpected, it pro vides a useful and necessary check of our methods. F ollo wing, the remaining 16, non-secular resonant contrib utors to the third order SVFS susceptibility will be gi v en. Ev en with the el imination of tw o thirds the terms, the follo wing deri v ati on is tedious, and the present approach is dif ferent than the more common Loui ville space approach. [5] T o mak e a connection between the present method [6, 6, 50, 87] and alternati v e approaches, an equi v al ent deri v ation of the well kno wn correspondence between third order optical K err ef fect (OKE) spectroscop y and the linear Raman e xperiment is gi v en in Chapter 6. This chapter sho ws ho w the third order OKE signal is, lik e the linear Raman measurement, determined by the autocorrelation of the system' s polarizability Chapter 6 also serv es to 59

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clarify the dif ferent approximations in v olv ed when resonant and of f-resonant elds are present. Equation 5.2 denes the third order polarization, P (3) k in the limit of monochromatic elds and in terms of its' Cartesian tensor components. P (3) k = (3)k j ih E j ( 1 ) E i ( 2 ) E h ( 3 ) (5.2) F or the purposes of describing third order SVFS, 1 2 and 3 are chosen to represent the visible eld, infrared eld, and static eld respecti v ely Further s = 1 + 2 + 3 and is the signal frequenc y while S V F S = 1 + 2 and is the sum frequenc y eld. In SVFS spectroscop y it is only the infrared eld that is resonant; both the signal and visible eld f all f ar from resonance. Because the third eld is static 3 0 and S V F S = s Gi v en the coupling of the infrared and visible elds to a static eld, the r esonant portion of the third order susceptibility tensor is gi v en by : (3) ;R E S k j ih = X g m nv g g ~ 3 ( (1 a ) ig v jv n knm hmg ( nm + s + ir nm )( mv 2 ir mv )( v g + 2 + ir v g ) +( 1 b ) ig v hv n jnm kmg ( v g + 2 + ir v g )( ng + 2 + ir ng )( mg + s + ir mg ) +( 2 a ) kg v jv n hnm img ( v g s ir v g )( ng 2 ir ng )( mg 2 ir mg ) +( 2 b ) hg v jv n knm img ( nm + s + ir v g )( v m + 2 + ir ng )( mg 2 ir mg ) +( 3 a ) ig v kv n jnm hmg ( nv s ir nv )( mv 2 ir mv )( v g + 2 + ir v g ) 60

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+( 3 b ) hg v kv n jnm img ( nv s ir nv )( v m + 2 + ir v m )( mg 2 ir mg ) +( 4 a ) ig v hv n knm jmg ( mn s ir mn )( ng + 2 + ir ng )( v g + 2 + ir v g ) +( 4 b ) jg v kv n hnm img ( v n + s + ir v n )( ng 2 ir ng )( mg 2 ir mg ) +( 5 a ) ig v hv n knm jmg ( nm + s + ir nm )( mv s ir mv )( v g + 2 + ir v g ) +( 5 b ) ig v jv n hnm kmg ( ng + s + ir ng )( mg + s + ir mg )( v g + 2 + ir mg ) +( 6 a ) ig v kv n hnm jmg ( nv s + ir nv )( mv s ir mv )( v g + 2 + ir v g ) +( 6 b ) ig v jv n knm hmg ( ng + s + ir ng )( mn s ir mn )( v g + 2 + ir v g ) +( 7 a ) kg v hv n jnm img ( ng s ir ng )( v g s ir v g )( mg 2 ir mg ) +( 7 b ) jg v hv n knm img ( nm + s + ir nm )( v m + s + ir v m )( mg 2 ir mg ) +( 8 a ) jg v kv n hnm img ( nv s ir v g )( v m + s + ir v m )( mg 2 ir mg ) +( 8 b ) hg v kv n jnm img ( v n + s + ir v n )( ng s ir ng )( mg 2 ir mg ) ) (5.3) Here, the alphanumeric labels in parenthesis enumerate subsets of terms within Equation 5.3. Notice there are doubly and singly resonant terms in Equation 5.3. The doubly resonant terms are identied as those that contain tw o f actors of 2 in the denominator The doubly resonant terms describe a resonant enhancement coupling to the static eld and then to the visible eld. The singly resonant terms describe the sum frequenc y gener ated eld coupling to the static eld. (This ordering of the elds can more easily be seen by representing the frequenc y domain perturbati v e e xpressions as their F ourier -Laplace transform of their corresponding time-dependent e xpression.) Gi v en, the frequenc y do61

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main e xpression in Equation 5.3, one can no w proceed to simplify the e xpression by grouping sets of terms to form polarizabilities, and subsequently transforming the resulting TCF e xpressions into the time domain. 5.3 A TCF A ppr oach to Quantify (3) ;R E S Contrib utions Be ginning from the microscopic e xpression in Equation 5.3, a TCF approach for calculating third order contrib utions to the SVFS signal is deri v ed that describes the third order ef fects due to the presence of a static eld at the interf ace. In de v eloping these correlation functions, we are guided by the requirement that the response function in time, i.e the F ourier transform of the frequenc y domain susceptibility must be real. [6, 90] This implies the nal quantity must be either the real or imaginary portion of t he comple x correlation function – depending on whether the pref actors are purely real or imaginary [89, 90] Thus, a minimum of tw o correlation functions is required such that their sum or dif ference necessarily cancels the real or imaginary component of the comple x correlation functions that result from grouping the terms in Equation 5.3 and subsequently F ourier transforming to the time domain. Note, in the rotating w a v e approximation that is commonly in v ok ed in the literature, the response is comple x [2, 6] and only agrees with the e xact e xpression in the high frequenc y limit. [6] T o proceed, it is helpful to restate the follo wing identities and denitions that are used in deri ving the ne w TCF' s: [1, 34] i Z 1 0 dt e it ( a ib ) = 1 a bi (5.4) 62

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i Z 1 0 dt e it ( a ib ) i Z 1 0 dt 0 e it 0 ( a ib ) = 1 ( a bi ) 2 (5.5) ij ( w ) = X g n ig n jg n ( ng w ir ng ) + jg n ig n ( ng + w + ir ng ) (5.6) A ( t ) = e iH t ~ Ae iH t ~ (5.7) ij k mg ( ) = 1 ~ X n ij mn kng ng ir ng + ij ng kmn ng + + ir ng # (5.8) Equation 5.8 should als o be f amil iar to the reader It describes the frequenc y dependent h yperpolarizability , deri v ed from the second order polarizability tensor specically for the case of SVFS. [6, 50, 52] First, the terms in Equation 5.3 that ha v e denominators that are doubly dependent on the resonant frequenc y ( 2 = I R ) will be e xamined. Combining 2a with 4b: C ( 2 = I R ) = 2 a + 4 b = 1 ~ 3 X ng mv g g img hv m ( v g 2 ir v g )( mg 2 ir mg ) kg n jnv ( ng s ir ng ) + knv jg n ( nv + s + ir nv ) # (5.9) C ( 2 = I R ) = 1 ~ 2 X v mg g g k j g v hv m img ( v g 2 ir v g )( mg 2 ir mg ) (5.10) Utilizing the inte gral identity in Equation 5.5 and performing the sum o v er states, Equation 5.10 can be written as the double half-sided (F ourier -Laplace) transform of a no v el TCF: C ( 2 = I R ) = 1 ~ 2 Z 1 0 dt e i! 2 t Z 1 0 dt 0 e i! 2 t 0 < k j ( t + t 0 ) h ( t ) i (0 ) > (5.11) 63

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Introduction of the (of f-resonant) static polarizability , in Equation 5.10 is not e xact. It essentially restricts the inde x v to the vibrational le v els in the electronic ground state. This is an e xcellent approximation under typical thermal conditions. Equation also assumes that the visible and sum frequenc y elds are not resonant with an electronic transition as the y may be in certain cases, [118] b ut not a typical SVFS e xperiment. Making the same approximations, it is possible to introduce the static polarizability by combination of the remaining doubly resonant terms (1a with 3a, 1b with 4a, and 2b with 3b). The results are gi v en belo w A ( 2 = I R ) = 1 a + 3 a = 1 ~ 2 X mv g g g ig v hmg k j v m ( mv 2 ir mv )( v g + 2 + ir v g ) = 1 ~ 2 Z 1 0 dt e i! 2 t Z 1 0 dt 0 e i! 2 t 0 < i (0 ) k j ( t + t 0 ) h ( t ) > (5.12) C ( 2 = I R ) = 1 b + 4 a = 1 ~ 2 X nv g g g ig v hv n k j ng ( ng + 2 + ir ng )( v g + 2 + ir v g ) = 1 ~ 2 Z 1 0 dt e i! 2 t Z 1 0 dt 0 e i! 2 t 0 < i (0 ) h ( t ) k j ( t + t 0 ) > (5.13) A ( 2 = I R ) = 2 b + 3 b = 1 ~ 2 X mv g g g img hg v k j v m ( mg 2 ir mg )( v m 2 ir v m ) = 1 ~ 2 Z 1 0 dt e i! 2 t Z 1 0 dt 0 e i! 2 t 0 < h ( t ) k j ( t + t 0 ) i (0 ) > (5.14) Note the TCF' s in Equations 5.12 – 5.14 and 5.3 are equi v alent classically and represent dif ferent ordering of the quantum mechanical operators. Collecti v ely the doubly resonant portion of (3) ;R E S k j ih is then gi v en as a sum of four terms: 64

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(3) ;D R E S k j ih = Z 1 0 Z 1 0 dt dt 0 e i! 2 ( t 0 + t ) f C ( t; t 0 ) + C ( t; t 0 ) A ( t; t 0 ) A ( t; t 0 ) g (5.15) Notice A ( t; t 0 ) and A ( t; t 0 ) and C ( t; t 0 ) and C ( t; t 0 ) are comple x conjug ates respecti v ely in time; thei r real and imaginary parts are denoted by subscript R and I respecti v ely Denoting F T as the full F ourier trans form: F T [ A ( t; t 0 ) = A R ( t; t 0 ) + iA I ( t; t 0 )] = A R ( ; 0 ) + A I ( ; 0 ) i.e their F ourier transforms are real due to the time symmetries of the real and imaginary parts of the TCF' s. Expanding the correlation functions into their real and imaginary components f acilitates the re writing of Equation 5.15 in terms of the real part of tw o dif ferent quantum mechanical TCF' s. (Note, the leading ~ dependence of the e xpression is no w sho wn e xplicitly dening the TCF' s as, e .g ~ A = ~ 2 A The tilde TCF' s are used for emphasis when the ~ dependence is informati v e.): (3) ;D R E S k j ih ( I R ) = 2 ~ 2 Z 1 0 dt e i! I R t Z 1 0 dt 0 e i! I R t 0 [ ~ A R ( t; t 0 ) ~ C R ( t; t 0 )] (5.16) Equation 5.16 contains a f actor of 1 ~ 2 thus the ~ 0 limit is not straightforw ard. Fur ther ~ A R ( t; t 0 ) and ~ C R ( t; t 0 ) are classically equi v alent b ut must dif fer starting at order ~ 2 for a well dened classical limit to e xist. [17, 36] T o proceed, a relationship between the quantum mechanical TCF' s A and C must be established. Writing TCF' s A ( t; t 0 ) and C ( t; t 0 ) in their Heisenber g representation and F ourier transforming to the frequenc y domain gi v es: A ( ; 0 ) = 1 ~ 2 X nv g e E g Q p ig v k j v n hng ( v g ) ( v n 0 ) (5.17) 65

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C ( ; 0 ) = 1 ~ 2 X nv g e E g Q p ig v hv n k j ng ( v g + ) ( ng + 0 ) (5.18) In Equations 5.17 and 5.18, Q p is the partition function. Switch dummy indices in Equation 5.18, and making use of symmetric nature of the real and matrix elements, the frequenc y domain e xpressions of A and C can be related via detailed balance by a single frequenc y v ariable: A ( ; 0 ) = e ~ C ( ; 0 ) (5.19) Equation 5.19 permits re writing of Equation 5.16 in terms of a single TCF ~ C : (3) ;D R E S k j ih ( I R ) = 1 ~ 2 Z 1 0 dt e i! I R t Z 1 0 dt 0 e i! I R t 0 Z 1 1 d! e i! t Z 1 1 d! 0 e i! 0 t 0 h ( e ~ 1) ~ C ( ; 0 ) + ( e ~ 1) ~ C ( ; 0 ) i (5.20) In the classical limit, the e xponentials in Equation 5.20 can be T aylor e xpanded. This cancels a f actor of ~ and allo ws Equation 5.20 to be written in terms of only the imaginary part of TCF ~ C Using classical si mulation techniques, only the (classical limit of the) real part of a comple x quantum mechanical TCF can be calculated. Thus, it is necessary to est ablish a re lationship between the real and imaginary components of ~ C ( ; 0 ) that w ould produce another f actor of ~ in the classical limit. [36] Considering ~ C ( ; 0 ) there is no e xact relationship between the real and imaginary parts of ~ C Ho we v er in the case of a strictly time independent static eld things simplify considerably Wri ting out ~ C ( ; 0 ) in t he ener gy representation, it can be sho wn that the three point correlation function reduces to a tw o point correlation function because matrix elements of the static dipole ij = h i j j j i = ij and colla pses one of the sums. All 66

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tw o point correlation functions ha v e real and imaginary parts that are analytically related by a frequenc y f actor of tanh ; in this case, the real and i maginary components of TCF ~ C are related in frequenc y space by: ~ C I ( ) = tanh ( ~ = 2) ~ C R ( ) [119] Expanding correlation function ~ C in Equation 5.16 into its real and imaginary components, and using this relationshi p to write the sum of correlation functio ns in terms of only the real part, results in e xact cancellation of all com ponents. Thus, to a good rst approximation, while the pathw ays in v olving the double resonance represent a distinct ph ysical process, their response is not observ ed due to total deconstructi v e interference. In a real system, an intrinsic static eld, due to char ged species at the interf ace, will uctuate in time about its a v erage v alue. In that case, one must resort to relating the real and imaginary parts of ~ C for a model system [17, 36] and obtain a general e xpression in terms of ~ C R (appropriate for all frequencies when quantum corrected) that also has a well dened classical limit. Considering a model harmonic system with the dipole and polarizability e xpanded out to second order in the harmonic coordinat e Q (one might also consider an anhar monic oscillator with a linear dipole and polarizability b ut this contrib ution is e xpected to be smaller), [17, 34] where primed quantities represent deri v ati v es with respect to the coordinate: ij = o ij + 0 Q ij + 00 Q 2ij 2 (5.21) ; ij = o ; ij + 0 ; Q ij + 00 ; Q 2ij 2 (5.22) 67

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Q ij = h i j Q j j i = ~ 2 m n 1 = 2 h p j i;j 1 + p j + 1 i;j +1 i (5.23) Q 2ij = ~ m n h ( j + 1 = 2) ij + p j ( j 1) i;j 2 + p ( j + 1 )( j + 2 ) i;j +2 i (5.24) The resulting higher order (non-stati c) terms of this TCF are isomorphic to the 5 th order Raman case [36] and similar considerations produce a TCF e xpression to describe this ef fect. Note, this contrib ution is only inseparable from the SVFS signal for components of the “static” eld that are slo wly v arying in time for which k IR + k vis + k st ati c k IR + k vis In this case, a contrib ution from this TCF w ould be possible. Ne xt, the remaining singly resonant terms need to be considered. Before introduction of the polarizability is pos sible, simplication of terms 6a and 7b (5a and 8a) is necessary and can be accomplished through combination of each with tw o (one) of f-resonant terms – making the same approximation ( v is s ) used in deri ving the normal resonant second order susceptibility correlation function for SVFS. [6] T erm s 5a, 6a, 7b, and 8a can then be re written as: 5 a = 1 ~ 3 X ng mv g g ig v hv n knm jmg ( nm + s + ir nm )( mg v is ir mg )( v g + I R + ir v g ) (5.25) 6 a = 1 ~ 3 X ng mv g g ig v kv n hnm jmg ( mg v is ir mg )( ng v is ir ng )( v g + I R + ir v g ) (5.26) 7 b = 1 ~ 3 X ng mv g g jg v hv n knm img ( mg I R ir mg )( ng + v is + ir ng )( v g + v is + ir v g ) (5.27) 8 a = 1 ~ 3 X ng mv g g jg v kv n hnm img ( mg I R ir mg )( v g + v is + ir v g )( nv s ir nv ) (5.28) 68

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Combination of 5a with 6a, 5b with 6b, 7b with 8a, and 7a with 8b allo ws for introduction of the non-resonant polarizability and can no w be written respecti v ely as: 5 a + 6 a = 1 ~ 2 X g mv g g ig m k h mv jv g ( v g v is ir v g )( mg + I R + ir mg ) (5.29) 5 b + 6 b = 1 ~ 2 X g m v g g ig m jmv k h v g ( v g + s + ir v g )( mg + I R + ir mg ) (5.30) 7 b + 8 a = 1 ~ 2 X g m v g g jg v k h v m img ( mg I R ir mg )( v g + v is + ir v g ) (5.31) 7 a + 8 b = 1 ~ 2 X g m v g g k h g m jmv img ( v g s ir v g )( mg I R ir mg ) (5.32) In forming Equations 5.29 – 5.32 we ha v e ag ain restricted v to the vibrational le v els of the electronic ground state. Usin g the denition of the h yperpolarizability in Equation 5.8, and the inte gral identity in Equation 5.4, the sum of singly resona nt terms can be written as: (3) ;R E S 0 k j ih ( I R ) = 2 ~ Z 1 0 dt e i! I R t ~ G I ( t ) (5.33) ~ G ( t ) = ~ G ( t ) = h k hj ( t ) i (0 ) i (5.34) It can be sho wn through detailed balanc e analysis that the F ourier transform of the real, G R ( t ) and imaginary G I ( t ) portions of the TCF are analytically related by a frequenc y f actor: G I ( ) = tanh ( ~ = 2) G R ( ) (Ag ain, where G R ( ) and G I ( ) are both real.) Thus, this correlation function has a well dened classical limit. 69

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(3) ;R E S 0 k j ih ( I R ) = 2 ~ Z 1 0 dt e i! I R t ~ G I ( t ) = 2 ~ Z 1 0 dt e i! I R t Z 1 1 d! e i! t ta nh ~ 2 ~ G R ( ) ) l im cl a ssical = Z 1 0 dt e i! I R t Z 1 1 d! e i! t G C l ( ) (5.35) In Equation 5.35, G C l is the classical TCF of the system' s uctuating h yperpolarizability and system dipole. Note, in applying the resulting TCF theory one w ould proceed to calculate the classical limit of G R ( t ) tak e its F ourier transform, and subsequently quantum correct the result. Constructing a quantum correction scheme is straightfor w ard – as described in our pre vious w ork, b ut typically does no t greatly change the lineshapes. [17, 120] 5.4 Methods of Computational Implementation T o construct a spectroscopic model for the BSM has been modied. Specically tensor h yperpolarizabilities can no w be assigned to each atom such that when the y interact, the y reproduce the equilibrium g as phase h yperpolarizability tensor for a gi v en molecule of interest. [47, 91] This ne w spectroscopic model is referred to as NSM. The h yperpolarizabilities and their deri v ati v es (dependence on molecular geometry is included e xplicitly in the ne w spectroscopic model) can be determined from ts to ES calculations using a n appropriate basis set. [121–125] The e xtended Applequist/Thole lik e model [6, 19, 38, 43–45, 45, 52, 126–130] that has been constructed gi v es the ef fecti v e h yperpolarizability (intrinsic + interaction ef fects), ef f ij k as sums o v er the products of the 70

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isolated and condensed phase polarizabil ity matrix elements of the system. The entire process is analogous to the Applequist/Thole man y-body polarization method we use to calculate the pola rizability [47] The computational ef fort is not signicantly increased because the most demanding step is still the iteration of the man y-body polarization equations. W ithin this formalism the total ef fecti v e h yperpolarizability (intrinsic + interaction ef fects), ef f ij k is gi v en by: [34, 47] ef f ; r ij k = X n n ef f ; ni n ef f ; nj n ef f ; r nk n (5.36) In Equation 5.36, n is the intrinsi c h yperpolarizability tensor associated with atom n ef f ; ni is the total ef fecti v e polarizability tensor component between atom i and n and n is the intrinsic point polarizability associated with atom n [47] The NSM w as implemented to de v elop a model of the system h yperpolarizability for w ater T o our kno wledge, only tw o e xperiments [114, 115] that were published in the 1960' s ha v e in v estig ated the h yperpolarizability of g aseous w ater Gi v en that technological inno v ations since that time ha v e led to more accurate data in terms of measuring g as phase elect ric moments, the v alues obtained from the referenced w orks were used as a rough guide only All parametrization for the h yperpolarizability of w ater w as performed using detailed ES calculations with PCGAMESS, and also by comparing to other theoretical in v estig ations. [131] The equi librium w ater congurations along with the polarizability parameters used in the ES and NSM calculations are those detailed pre viously in Chapter 3. The ES calcula71

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tions were done at the aug-cc-pvqz le v el, and electron correlation w as accounted for using B3L YP5. (MP2, used in pre vious BSM de v elopments, is a v ali d option for electron correlation when calculating polariza bilities b ut not h yperpolarizabilities in PCGAMESS. F or consistenc y it w as check ed to ensure that B3L YP5 and MP2 produced the same polarizability deri v ati v es. The y do.) The h yperpolar izability deri v ati v e tensors were obtained via the 9 coordinate displacement method detailed in Chapter 3. First the intrinsic h yperpolarizability tensors of Oxygen and Hydrogen were deter mined such that when the y inte racted via the NSM, the g as phase h yperpolari zability tensor of w ater calculated using PCGAMESS, w as reproduced. The best t intrinsic h yperpolarizability tensors parameters associated with the Oxygen and Hydrogen are presented in T able 5.1. T able 5.1 also compares the interaction h yperpolarizability tensor of a single w ater molecule computed via ES and NSM methods. Agreement between the tw o methods is remarkable – especially considering pre vious techniques that are capable of calculating the h yperpolarizability of w ater on the y were only able to match ES calculations within a 15% error [91] Subsequently the intrinsic h yperpolarizability deri v ati v e tensors of Oxygen and Hydrogen were parametrized. Introduction of the h yperpolarizabili ty deri v ati v e tensors allo ws the total h yperpolarizability t o be e xpressed as a function of both displacement from equilibrium bond length ( r ) an d angle ( ): ij k = ij k o + 0 r ij k r + 0 ij k Extensi v e global scan methods were emplo yed in pursuit of nding a composition of the h yperpolar izability deri v ati v e tensors for Oxygen and Hydrogen that w ould model the uctuations 72

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xxz xxz y y z y y z z z z O 0.679 -0.271 0.059 H -0.6725 0.6875 -0.8625 ES -0.5733135 -0.2205550 -0.6087189 NSM -0.5733011 -0.2205574 -0.6087125 %Error -0.0021628655 0.0010881640 -0.0010513884 T able 5.1: The non-zero intrinsic h yperpolarizability te nsor parameters of Oxygen and Hydrogen used in the NSM calculations are gi v e bel o w F ollo wing, the non-zero inter action h yperpolarizability tensor elements of an equilibrium conguration isolated w ater molecule are gi v en for both ES and NSM methods along with their percent error Note, symbolizes the h yperpolarizability is fully symmetric with respect to all superscripts. 73

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0 r xxz 0 y y z r 0 r z z z O 6.300 -0.650 5.260 H 3.75 37.95 -0.5 0 xxz 0 y y z 0 z z z O 3.750 0. 3.0 H 8.933 0. -35.8 T able 5.2: The h yperpolarizability deri v ati v e parameters for the NSM of g aseous w ater are sho wn. Units are in A 4 =e for 0 r and A 5 /( e radian) for 0 of all ij k tensor components of the system b ut none where found. Ho we v er third order SVFS signals depend only on the system' s uctuations in the total h yperpolarizability in three specic directions: z x z xxz and z z z Further the most common SVFS measurement is tak en in the XXZ direction (because it gi v es by f ar the most intense signal). Therefore, while not ideal, it is suf cient to ha v e a model that captures the system' s uctuating h yperpolarizability deri v ati v e tensor in the XXZ direction. T able 5.2 gi v es the non-zero h yperpolarizability deri v ati v e tensor parameters for Oxygen and Hydrogen that best reproduced the interaction h yperpolarizability tensor of g aseous w ater There were se v eral global ts that reproduced the XXZ component of the interaction h yperpolarizability deri v ati v e tensor The set of parameters that w as selected w as chosen based upon its ability to also reproduce the tw o other rele v ant directions (ZXZ and ZZZ) in the majority of coordinate displacements. 74

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T able 5.3 lists the resultant interaction h yperpolarizability deri v ati v e tensor in the XXZ, ZXZ, and ZZX directions calculated using ES and NSM methods. Note, only the deri v ati v es that ha v e an order of magnitude E 1 A 4 =e in the SVFS rele v ant directions are gi v en in T able 5.3. In pursuit of further testing the NSM, the total h yperpolar izability of a w ater dimer w as calcul ated using both ES and NSM methods. The same coordinate displacement method, detailed pre viously w as implemented to obtain the polarizability matrix. In this case, 18 displacements were made instead of 9. The results of ES and NSM methods were compared to determine whether the NSM w as capable of capturing the inuence of the additional interactions af fect on the system h yperpolarizability The XXZ h yperpolarizability deri v ati v e tensor element calculated via NSM e xhibits at most a 2.4% dif ference from the ES determinations, and this dif fere nce is as small as .16%. The ZZZ and ZZX elements are not reproduced as well as the XXZ element. In general, most of the ZZZ and ZZX tensor elements are within a percent dif ference of 2-3%. Ho we v er there are outliers that f all f ar from the ES v alue – as high as 139% dif ference in the ZZX direction. The model w as further tested by assessing its ability to capture the h yperpolarizability tensor components of condensed phase w ater as compared to tw o other detailed, ES based, theoretical in v estig ations. [91, 124] The system congurations were generated using CM3D, [132] an MD code de v eloped at the Uni v ersity of Pennsylv ania. Microcanonical MD of 64 w ater molecules w as performed at a density of 1.0 g /cm 3 and an a v erage temperature of 298K. 75

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Displacement Component ES NSM %Dif ference Ox: 0 z x z 0.535820 0.536980 0.11598 Oz: 0 xxz 1.371292 1.382532 -0.23294 Oz: 0 z z z 0.866077 0.863747 1.123897 H1x: 0 xxz -1.34889 -1.34932 -0.0433224 H1x: 0 z z z -0.491203 -0.491037 0.016639 H1x: 0 z z x -0.267954 -0.268592 -0.063821 H1z: 0 xxz -0.685743 -0.691192 -0.54485 H1z: 0 z z z -0.433137 -0.8431843 0.129398 H1z: 0 z z x -0.5627319 -2.002802 -144.007 H2x: 0 xxz 1.348770 1.349364 0.059449 H2x: 0 z z z 0.491239 0.49072155 -0.05070 H2x: 0 z z x -0.26790 -0.268388 -0.04881 H2z: 0 xxz -0.68574 -0.6911921 -0.54514 H2z: 0 z z z -0.433117 -0.4318426 0.127441 H2z: 0 z z x 0.562732 2.002801 144.007 T able 5.3: The h yperpolarizabilit y deri v ati v e components of g aseous w ater that are rele v ant to third order SVFS and non-zero for a gi v en displacement are listed for both ES and NSM calculation methods. The percent error between NSM and ES methods is also listed. Note, the rst column designates the coordinate that w as displaced as described in the te xt, and units are in A 4 =e 76

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The XXZ and YYZ components of the h yperpolarizability tensor compare well with the other ES based theoretical in v estig ations. Specically the change of sign that is associated with the h yperpolarizability tensor of w ater going from the g as phase to the condensed phase is reproduced! Our method calculates X X Z =0.18 a.u. and Y Y Z =10.60 a.u.. V alues calculated by Jensen (K usalik et. al) using v arious ES techniques f all in the range of X X Z =0.14-0.81 (4.1-5.7) a.u. and Y Y Z =7.5-9.03 (10.9-18.8) a.u.. The Z Z Z component calculated by our model is ob viously incorrect – the trademark change of sign is not present in this direction. Our model calculates Z Z Z =-42.175 a.u.. Despite it s hort comings, the NSM does rema rkably well in reproducing the most rele v ant component to third order SVFS – the XXZ component of the s ystem h yperpolarizability deri v ati v e tensor for a w ater a w ater dimer and condensed phase w ater F or calculations in v olving one and tw o w ater molecules, the total system h yperpolarizability calculated via ES and NSM agree well. F or calculations in v olving condensed phase w ater only tw o of the three non-zero h yperpolarizability tensor components could be reproduced. T o date, no other molecularly detailed, on the y technique has been able to do as well in reproducing a system' s h yperpolarizability let alone the h yperpolarizability deri v ati v es and for a system as complicated as w ater! Further in v estig ation and renement of this model is clearly required, b ut the initial results are promising. Specically because the total system h yperpolari zability is a function of intrinsic and system polarizability elements, it' s advisable that the NSM be modi ed to incorporate an atom' s intrinsic polarizability tensor instead of this quantity being approximated as a point polarizability 77

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Chapter 6 Linear Raman: A Fr equency-T ime Deri v ation of the Response Function Of f-resonant Raman e xperiments emplo y three indi vidually of f-resonant visible elds. [5] The w a v e v ector matching condition and the signal freque nc y are k s = k 1 k 2 + k 3 and s = 1 2 + 3 respecti v ely The resonant frequenc y is at 1 2 and the signal frequenc y is of f-resonant. In some of f-resonant Raman e xperiments, k 1 = k 3 This is not a necessary condition, and thus we treat k 1 and k 3 to be distinguishable elds. Starting from corrections to the Schrodinger w a v efunction formalism, since no individual eld is resonant, [2, 133] and a v eraging o v er the initial distr ib ution of states, when all three elds are distinguishable, there are 24 terms tha t describe the third order susceptibility F or of f-resonant Raman, only terms that contain a denominator of the form ( ab 1 2 ir ab ) will contrib ute to the resonant polarization. Thus, only eight of the twenty four terms need to be considered when determining the resonant susceptibility Starting from Equation 5.2 as the denition of t he polarization in terms of its' Cartesian tensor components, and assuming the three elds are all distinguishable, the r esonant portion of the susceptibility is gi v en by: 78

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(3)k j ih ( s ; 1 ; 2 ; 3 ) = X g m nv g g ~ 3 +( 1 d ) kg v hv n inm jmg ( v g s ir v g )( ng 1 + 2 ir ng )( mg 1 ir mg ) +( 1 e ) kg v hv n jnm img ( v g s ir v g )( ng 1 + 2 ir ng )( mg + 2 ir mg ) +( 2 d ) hg v kv n inm jmg ( v g + 3 + ir v g )( ng 1 + 2 ir ng )( mg 1 ir mg ) +( 2 e ) hg v kv n jnm img ( v g + 3 + ir v g )( ng + 2 1 ir ng )( mg + 2 ir mg ) +( 3 a ) jg v iv n knm hmg ( v g + 1 + ir v g )( ng + 1 2 + ir ng )( mg 3 ir mg ) +( 3 b ) ig v jv n knm hmg ( v g 2 + ir v g )( ng + 1 2 + ir ng )( mg 3 ir mg ) +( 4 a ) jg v iv n hnm kmg ( v g + 1 + ir v g )( ng + 1 2 + ir ng )( mg + s + ir mg ) +( 4 b ) ig v jv n hnm kmg ( v g 2 + ir v g )( ng + 1 2 + ir ng )( mg + s + ir mg ) (6.1) In the follo wing algebraic manipulation, we introduce the static polarizability , because all three elds and the signal eld f all f ar from resonance. F ormally introduction of the static polarizability amounts to approximating 1 2 3 s : 1 d + 2 d = 1 ~ 3 X mng v inm jmg ( ng 1 + 2 ir ng )( mg 1 ir mg ) 79

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" kg v hv n v g w s ir v g + hg v kv n v g + w 3 + ir v g # = 1 ~ 2 X mng inm jmg k h g n ( ng 1 + 2 ir ng )( mg 1 ir mg ) (6.2) 1 e + 2 e = 1 ~ 3 X mng v jnm img ( ng 1 + 2 ir ng )( mg + 2 + ir mg ) kg v hv n v g w s ir v g + hg v kv n v g + w 3 + ir v g # = 1 ~ 2 X mng jnm img k h g n ( ng 1 + 2 ir ng )( mg + 2 + ir mg ) (6.3) 3 a + 4 a = 1 ~ 3 X mng v imn jg m ( ng + 1 2 + ir ng )( mg + 1 + ir mg ) (6.4) knv hv g v g w 3 ir v g + hnv kv g v g + w s + ir v g # = 1 ~ 2 X mng imn jg m k h ng ( ng + 1 2 + ir ng )( mg + 1 + ir mg ) (6.5) 3 b + 4 b = 1 ~ 3 X mng v ig m jmn ( ng + 1 2 + ir ng )( mg 2 + ir mg ) knv hv g v g w 3 ir v g + hnv kv g v g + w s + ir v g # = 1 ~ 2 X mng ig m jmn k h ng ( ng + 1 2 + ir ng )( mg 2 + ir mg ) (6.6) 80

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Combining Equations 6.2 with 6.3 and 6.5 wit h 6.6 result s in Equat ions 6.7 and 6.8 respecti v ely E q : 6 : 2 + E q : 6 : 3 = i ~ Z 1 0 dt e it ( 2 1 ) < k h ( t ) ij (0 ) > (6.7) E q : 6 : 5 + E q : 6 : 6 = i ~ Z 1 0 dt e it ( 2 1 ) < k h (0 ) ij ( t ) > (6.8) Equation 6.8 is the comple x conjug ate of 6.7. Their sum is, thus, equal to twice the imaginary portion of the correlation function. Making use of this simplication, the time domain Raman susceptibility tensor (3) ;R A M AN k j ih ( t ) is no w gi v en by: (3) ;R A M AN k j ih ( t ) = 2 ~ I m < k h ( t ) ij (0 ) > (6.9) The imaginary portion of the polarizability autocorrelation function, can be related via the uctuation dissipation theorem to the real part, and the real part to its' classical analog. [5] 81

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Chapter 7 Conclusion Spectroscop y is a po wer tool for in v estig ating the structure and dynamics of chemical systems. A system' s spectroscopic response (susceptibility) is formally described by a comple x quantum mechanical quantity which appears to ha v e no ob vious classical limit. This manuscript has presented mathematical techniques applied to general susceptibility e xpressions that allo w for unique TCF' s describing a specic optical process to be deri v ed. These TCF' s are sho wn to ha v e a deniti v e classical limit, and are thus, amenable to semiclassical computation techniques. The semiclassical computation technique de v eloped and used in this w ork hinges upon tw o primary components – c lassical MD and a spectroscopic model. Specically the trajectories generated from classical MD are used as input into the spectroscopic model such that the uctuations in t he system' s electric moments and electric moment' s deri v ati v es are captured as accurately as possible. Thus, the TCF' s that descr ibe a particular type of spectroscop y are able to also be calculated as accurately as possible. 82

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The rob ust nature of our semi classical computat ion technique in calculating spectroscopic observ ables in a molecularly detailed manner has also been tested. Modeling the spectroscop y of liquid w ater systems in a molecularly detailed methodology has histor ically been considered dif cult [42, 50, 82, 134–136] due to the strong man y-body inter actions that are present. The spectroscopic model presented in this m anuscript is able to capt ure the nature of the comple x man y-body interactions as e videnced by it' s ability to reproduce the ne detail s (subpopulation and phase) of e xperimental interf acial w ater spectra. Further an e xtension of the spectroscopic model that goes be yond calculating the sys tem' s dipole and polarizability moments also has been presen ted. Preliminary calculations par ametrizing the h yperpolariz ability and h yperpolarizabil ity uctuations ha v e also been conducted for a w ater m olecule and a w ater dimer Initial results are in good agreement with other theoretical w ork, and appear to capture the total h yperpolarizability better than an y other molecularly detailed on the y method to date for a condens ed phase system. 83

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Refer ences [1] W Heitler The Quantum Theory of Radiati on (Do v er Publications Inc., Ne w Y ork, 1984). [2] R. W Bo yd, Nonlinear Optics (Academic Press, London, 2003). [3] Y Shen, Principles of Nonlinear Optics (W ile y Ne w Y ork, 1984). [4] P Butcher and D. Cotter The Elements of Nonlinear Optics (Cambridge Uni v ersity Press, Cambridge, 1990). [5] S. Mukamel, Principles of Nonlinear Optical Spectr oscopy (Oxford Uni v ersity Press, Oxford, 1995). [6] A. Perry C. Neipert B. Space, and P Moore, Chem. Re v 106 1234 (2006). [7] G. Richmond, Chem Re v 102 2693 (2002). [8] R. Re y K. B. Moller and J T Hynes, Chem. Re v 104 1915 (2004). [9] M. T Zanni, N.-H. Ge, Y Sam Kim, and R. M. Hochstrasser PN AS 98 11265 (2001). 84

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[10] P Hamm, M. Lim, W F DeGrado, and R. M. Hochstrasser J. Chem. Ph ys. 112 1907 (2000). [11] S. Li et al. J. Ph ys. Chem. B 110 18933 (2006). [12] A. N. Borden yuk and A. V Benders kii, J. Chem. Ph ys. 122 134713(1) (2005). [13] P Peterson and R Saykally Annu. Re v Ph ys. Chem. 57 333 (2006). [14] K. Eisenthal, C hem. Re v 106 1462 (2006). [15] E. Hao, G. Schatz, R. Johnson, and J. Hupp, J. Chem. Ph ys. 117 5961 (2006). [16] H. T orii, J. Ph ys. Chem A 110 9469 (2006). [17] R. DeV ane et al. J. Chem. Ph ys. 121 3688 (2004). [18] R. DeV ane, C. Ridle y T K e yes, and B. Space, J. Chem. Ph ys. 123 194507 (2005). [19] A. Perry et al. J. Chem. Ph ys. 123 144705(1) (2005). [20] N. Bloember gen and P Pershan, Ph ys. Re v 128 606 (1962). [21] N. Bloember gen, R. C hang, S. Jha, and C. Lee, Ph ys. Re v 174 813 (1968). [22] J. Armstrong, N. Bloember gen, J. Ducuing, and P Pe rshan, Ph ys. Re v 127 1918 (1962). [23] C. H. Lee, R. K. Chang, a nd N. Bloember gen, Ph ys. Re v Lett. 18 167 (1967). 85

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About the A uthor Christine L. Neipert majored in chemistry and minored in mathematics, recei ving a Bachelor' s of Science De gree from Maryville Uni v ersity in December of 2002. She entered the Doctoral program, studying computational ph ysical chemistry under Professor Brian Space, at the Uni v ersity of South Florida in January 2003. In addition to her research and formal course w ork, Ms. Nei pert attended and made presentations at se v eral national conferences and summer schools. She w as also one of nine nationally selected graduate students by Oak Ridge Associated Uni v ersities to attend the 2006 Chemistry Nobel Laureates and Students Conference in Lindau, German y While in the Ph.D. program at the Uni v ersity of South Florida, Ms. Neipert w as the recipient of the Latino Graduate Fello wship, Tharp Endo wed Scholarship, a Department of Ener gy tra v el grant, and the Gordon Research Conference Carl Storms tra v el grant. She w as also selected as a National Science F oundation Students-T eachers-and-Researchers graduate fello w and best oral presentation at the 2006 Castle Conference.


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Construction and application of computationally tractable theories on nonlinear spectroscopy
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ABSTRACT: Nonlinear optical processes probe systems in unique manners. The signals obtained from nonlinear spectroscopic experiments are often significantly different than more standard linear techniques, and their intricate nature can make it difficult to interpret the experimental results. Given the complexity of many nonlinear lineshapes, it is to the benefit of both the theoretical and experimental communities to have molecularly detailed computationally amenable theories of nonlinear spectroscopy. Development of such theories, bench marked by careful experimental investigations, have the ability to understand the origins of a given spectroscopic lineshape with atomistic resolution. With this goal in mind, this manuscript details the development of several novel theories of nonlinear surface specific spectroscopies. Spectroscopic responses are described by quantum mechanical quantities. This work shows how well defined classical limits of these expressions can be obtained, and unlike the formal quantum mechanical expressions, the derived expressions comprise a computationally tractable theory. Further, because the developed novel theories have a well defined classical limit, there is a quantum classical correspondence. Thus, semiclassical computational techniques can capture the true physics of the given nonlinear optical process. The semiclassical methodology presented in this manuscript consists of two primary components classical molecular dynamics and a spectroscopic model. For each theory of nonlinear spectroscopy that is developed, a computational implementation methodology is discussed and/or tested.
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