Theory of interfacial orientational relaxation ...

Theory of interfacial orientational relaxation spectroscopic observablesZsolt Gengeliczki, Daniel E. Rosenfeld, and M. D. Fayera

Department of Chemistry, Stanford University, Stanford, California 94305, USA

Received 20 January 2010; accepted 10 May 2010; published online 24 June 2010

The orientational correlation functions measured in the time-resolved second-harmonic generationTRSHG and time-resolved sum-frequency generation TRSFG experiments are derived. In thelaboratory coordinate system, the Yl

mlabtY2mlab0 l=1,3 and m=0,2 correlation

functions, where the Ylm are spherical harmonics, describe the orientational relaxation observables of

molecules at interfaces. A wobbling-in-a-cone model is used to evaluate the correlation functions.The theory demonstrates that the orientational relaxation diffusion constant is not directly obtainedfrom an experimental decay time in contrast to the situation for a bulk liquid. Model calculations ofthe correlation functions are presented to demonstrate how the diffusion constant and conehalf-angle affect the time-dependence of the signals in TRSHG and TRSFG experiments.Calculations for the TRSHG experiments on Coumarin C314 molecules at air-water andair-water-surfactant interfaces are presented and used to examine the implications of publishedexperimental results for these systems. © 2010 American Institute of Physics.doi:10.1063/1.3442446

I. INTRODUCTION

Interfaces and surfaces play an important role in hetero-geneous catalysis,1,2 electrochemistry,3 nanoscience,4 atmo-spheric chemistry,5,6 and biology.7,8 Therefore, it is importantto understand the special physical and chemical properties ofsuch interfaces at the molecular level. Static and dynamicmolecular properties of interfaces cannot be derived or esti-mated from bulk phase measurements, and techniques havebeen developed to determine the interfacial composition andorientation of molecules at interfaces.

Some of the most powerful tools for spectroscopic inves-tigation of interfaces are second-harmonic generation SHGand sum-frequency generation SFG spectroscopies.9,10

These techniques rely on the second-order nonlinear proper-ties of interfaces. In contrast to most bulk phases, interfacesare noncentrosymmetric. Because of this fact, SHG and SFGcan provide nearly interference-free measurements of inter-facial spectroscopic properties.11,12 One of the useful proper-ties measured with these techniques is the equilibrium orien-tation of molecules at interfaces and surfaces.11–14 SFG wasalso successfully used for in situ identification of reactionintermediates on nanoparticle catalysts both at atmosphericand high pressure.11–13,15–17

Although static measurements give invaluable insightsinto the structure of molecular interfaces, surface reactivityand transport phenomena cannot be described without under-standing of the dynamical behavior. Molecular orientationalrelaxation affects the SFG and SHG signals and limited in-formation can be obtained through the time-independentvariations in signal intensities with respect to electric fieldpolarizations.18 To obtain detailed dynamical information,time-dependent versions of SFG and SHG spectroscopieshave been developed in which either SFG or SHG are used

as probes subsequent to an incident pump pulse.19–23 Thesetechniques are typically referred to as time-resolved second-harmonic generation TRSHG and time-resolved sum-frequency generation TRSFG spectroscopies. In any time-resolved method such as TRSHG or TRSFG, molecularorientational relaxation contributes to the measured time-dependence of the signals.19,22,24,25 Such experiments areanalogous to measurements of orientational relaxation dy-namics of molecules in bulk liquids. The time-dependence ofthe interfacial molecular reorientation signal can provide in-formation on the nature of orientational motions of mol-ecules at an interface and permit comparison to the behaviorof molecules in bulk liquids. In recent years, measurementshave been made of the time scales of molecular reorientationand excited state population relaxation at the air-water inter-face with different pumping schemes.23–28 The variouspumping schemes typically differ in the polarization and fre-quency of the pump laser.

McGuire and Shen23 applied an infrared pump to the OHstretch of interfacial water molecules, which was subse-quently probed by SFG. Eisenthal and co-workers24,25 dem-onstrated that electronic excitation causes a depletion in boththe SHG and the SFG signals, and that reorientation of themolecules can be observed by following the recovery of thesignal. The time-dependent signal is caused by the reorienta-tion of the molecules at the interface and the electronic orvibrational lifetime relaxation. If the electronic or vibrationallifetime is long enough, the initial fast decay time constantcan be assigned to the reorientation of the interfacial mol-ecule. In many cases, the reorientational time constant iscompared to the time constant measured in the bulk phase.25

Eisenthal and co-workers24 have performed studies ofthe orientational relaxation of Coumarin 314 C314 at theair-water interface using both TRSHG and TRSFG experi-ments. They assigned the extracted time constants to theaElectronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 132, 244703 2010

0021-9606/2010/13224/244703/19/$30.00 © 2010 American Institute of Physics132, 244703-1

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

http://dx.doi.org/10.1063/1.3442446

http://dx.doi.org/10.1063/1.3442446

“in-plane” and “out-of-plane” reorientation of the moleculesat the interface. Their findings can be summarized in a fewpoints: 1 the orientational dynamics can be isolated fromthe population relaxation due to a separation of timescalesfor the coumarin system and 2 the SHG signal recoverycaused by the reorientational motion of the molecules issingle exponential and there is a remaining bleach once thereorientation is complete. In principle, the recovery of theSHG signal should be biexponential if both the in-planechange in the azimuthal angle and the out-of-plane changein the polar angle motions are observed. Eisenthal andco-workers27 stated that the two motions happen on a verysimilar time scale and they measure almost identical timeconstants. They directly compared the measured exponentialdecay time constants with the bulk phase orientational relax-ation time constants. As will be derived in detail below, sucha direct comparison is invalid due to a different relationshipbetween the decay times and the orientational diffusion con-stants. In contrast to bulk liquids, observed interfacial orien-tational decay times are not directly related to the inverse ofthe orientational diffusion constant.

Recently, Nienhuys and Bonn28 developed a numericalmethod for assigning rotational diffusion constants for thereorientation of molecules at the air-liquid interface studiedby TRSFG experiments. For the hypothetical situation inwhich orientational relaxation is only allowed in the azi-muthal angle a one-dimensional random walk with cyclicboundary condition, the dynamic nonlinear susceptibilitiesare expressed analytically as exponential decays. However,for the physically realistic situation in which reorientation isallowed in both the azimuthal and polar angles, a number ofspecific cases are considered and the results are reported asnumerical solutions. In their model, they assume a harmonicpotential that governs the “out-of-plane motion” change inthe polar angle of the molecules, while they treat the in-plane motion change in the azimuthal angle as a free rota-tion about the surface normal. The two types of motions areassumed to be decoupled, that is, the in-plane rotational dif-fusion constant does not depend on the out-of-plane orienta-tion angle. The in-plane and out-of-plane reorientational dy-namics can only be fully separated when the out-of-planeangle is restricted to a very narrow range.28 In addition, be-cause the framework for analyzing the out-of-plane motionin the TRSFG and TRSHG experimental data is numerical, itis difficult to obtain a broad understanding of the factors thatinfluence the experimental observables.

None of the above studies determine which correlationfunctions are measured in the TRSHG and TRSFG experi-ments. Without knowledge of the correlation functions asso-ciated with a particular experiment, experimental observ-ables cannot be calculated or simulated. Previously, therehave been no analytical treatments of the spectroscopic ob-servables of molecules undergoing orientational relaxation atinterfaces that relate the observables to the orientational dif-fusion constants.

In the present paper, we analyze polarization selectiveTRSFG and TRSHG experimental observables in a compre-hensive manner. First we will discuss which correlation func-tions are measured in these experiments. The electric-field-

matter interactions differ in number and type from those inbulk phase polarization selective pump-probe experiments,and they include both resonant and nonresonant interactions.The correlation functions will be obtained in general for awell defined molecular symmetry. Then the results are madespecific for a wobbling-in-a-cone model,29–31 in which theorientational motion is restricted to a cone of angles. Thewobbling-in-a-cone model allows us to express the time con-stant of the observed exponential decay as a function of thehalf-cone angle and to extract the orientational diffusion con-stant. The tilt angle of the cone relative to the interface nor-mal is also included in the treatment. This general tilt anglepermits the theory to treat cases in which the interfacial mol-ecules can diffuse within a range of angles that is not cen-tered on the surface normal. Model calculations are pre-sented to illustrate the theoretical results.

The theory presented below shows that the observed ori-entational decays for surface selective nonlinear experimentsare not directly related to the orientational diffusion constant.Therefore, the time constants of decays observed at inter-faces cannot be directly compared to the time constant ofdecays measured in the bulk. The results presented here notonly permit interfacial orientational diffusion constants to beobtained, but they also make it possible to extract informa-tion about the range of angles over which orientational re-laxation occurs from the time-dependent measurements.

Data from the literature on C314 at the air-water24,22 andair-water-surfactant25,27 interfaces are analyzed and reinter-preted. While the orientational relaxation time constants atthe air-water interface and in bulk water differ substantially,it is shown that the orientational diffusion constants are infact the same within a small error. In addition, although therelaxation time constants at the air-water interface and theair-water-surfactant interface are almost the same, the surfac-tant reduces the orientational diffusion constant by a factorof approximately three. These results demonstrate that it ispossible to extract detailed information on the influence ofinterfaces on molecular orientational dynamics by obtainingthe orientational diffusion constants from the data.

II. THEORETICAL FRAMEWORK

Since one of the goals of this work is to relate interfacialorientational relaxation correlation functions to those mea-sured for the bulk, we briefly review the observable for bulkorientational relaxation. Orientational dynamics of moleculesin the isotropic liquid phase can be studied with pump-probeand fluorescence anisotropy experiments. The theory of theseexperiments is well understood.32 The anisotropy of the ori-entational relaxation in the bulk phase is given by

rt =It − It

It + 2It=

2

5C2t , 1

where rt is the orientational anisotropy and It and Itare the components of the observable probe pulse and fluo-rescence measured parallel and perpendicular to the pumppulse, respectively.33,34 C2t is the second-order Legendrepolynomial orientational correlation function,35

244703-2 Gengeliczki, Rosenfeld, and Fayer J. Chem. Phys. 132, 244703 2010


C2t = P2t0 , 2

where is the transition dipole. For isotropic orientationaldiffusion of an ensemble of spherical rotors, the correlationfunction is a single exponential decay with time constant =1 /6Dor and rotational diffusion constant Dor, specifically

C2t = e−6Dort. 3

This is a well known formula that can be derived by solvingthe isotropic diffusion equation.36,37 Treatment has been pre-sented for other molecular shapes such as oblate and prolatespheroids.33,32

Molecules can also undergo restricted rotational diffu-sion if the local environment is anisotropic, for example, inglassy materials,31,38 macromolecules,39,40 membranes,41 andreverse micelles.42 A model that has been widely used todescribe these systems is wobbling-in-a-cone. The model in-volves a rigid rod with cylindrical symmetry whose cylindri-cal axis is confined within a conical volume, and the poten-tial is uniform within the cone and infinite outside of thecone. This is referred to as the hard-wall cone model. TheC2t correlation function can be expressed analytically andgives a single exponential decay to a nonzero constant. Themagnitude of the constant is determined by the cone half-angle. Combining the decay time constant and the cone angleyields the orientational diffusion constant for motion withinthe cone. Szabo33,43 presented comprehensive discussions ofthe wobbling-in-a-cone problem for probes embedded inmacromolecules, and macroscopically oriented samples,such as uniaxial liquid crystals.43 Pecora and Wang ex-pressed the orientational correlation function in terms ofspherical harmonics and gave a general and mathematicallyrigorous derivation of the Yl

mtYlm0 l=1,2; m

=0,1 ,2 correlation functions, which can be used as a tem-plate for employing the wobbling-in-a-cone-model in differ-ent types of experiments.30 The wobbling-in-a-cone modelhas been extended to a “harmonic cone” in which the cone isa two-dimensional parabolic potential.44

Below, the necessary formalism for solving for the ori-entational relaxation component of TRSHG probe andTRSFG probe experiments will be derived. As this derivationis long and intricate we will begin with a brief outline. First,in Sec. II A, the electric-field-matter interactions which giverise to pump-SHG probe and pump-SFG probe signals willbe formalized via diagrammatic perturbation theory. In Sec.II B, the tensorial nature of these interactions will be consid-ered, resulting in general expressions for the orientationalresponse functions. Given experimental conditions, the ori-entational correlation functions can be expanded in time cor-relation functions of spherical harmonics. The remainder ofthe work in Sec. II will be directed toward finding analyticalexpressions for the spherical harmonic correlation functionswhich contribute to the orientational response. Section II Cwill construct the angular coordinates for the problem andSec. II D will solve the model of restricted rotational diffu-sion in a cone. In Sec. III, the orientational response func-tions will be expanded in terms of spherical harmonics andsolved for in terms of the correlation functions derived inSec. II D. The general theory applied here is used to describe

the TRSHG and TRSFG observables in detail in Sec. III andmodel calculations and analysis of experimental data are pre-sented in Sec. IV.

A. Electric-field-matter interactions

The TRSHG and TRSFG experiments include the elec-tronic or vibrational excitation of the molecules under study,followed by a time-delayed second SHG probe or SFGprobe. The pump laser may be either an infrared laser or avisible laser and the probe may be either SHG or SFG. Com-bination of the pumping and probing schemes results in fourexperiments, three of which are likely to provide molecularinformation. These three experiments are visible-pump-SHGprobe, visible-pump-SFG probe, and infrared-pump-SFGprobe. The visible laser with frequency denoted vis is of-ten a near-IR laser with wavelength 800 nm. For performinga SHG probe experiment it is advantageous for this laser tobe two-photon resonant with an electronic excited state. Thisallows for efficient pumping via frequency-doubled laserlight and efficient probing via two-photon-resonance en-hancement. The IR laser should be resonant with a vibrationof interest; typical wavelengths are 3–5 m.

Both the pump-SHG probe and pump-SFG probe experi-ments are fourth-order nonlinear optical experiments, whichare described by tensorial nonlinear response functions.Here, the physical origin of the signals will be discussedwhile in Sec. II B, the tensorial nature will be addressed. Forsuch processes, it is useful to expand the response functionsusing diagrammatic perturbation theory.45,46 Within thistreatment, the total response function is found to consist of asum of multiple, independent, double-sided Feynman dia-grams. Each diagram represents the contribution of a timecorrelation function to the measured signal. Formally, eachtime correlation function, and therefore diagram, is a singleterm expanded from the nested commutator that results fromthe perturbation theory treatment.45 The full response func-tion also includes the complex conjugates of the diagramsshown.

Each double-sided Feynman diagram consists of a seriesof density matrix elements drawn vertically, one above theprevious. The axis of time increases from the bottom to thetop of the diagram. Each interaction is separated by a time n

as shown along side of Fig. 1a. The period between the twopump fields and the SHG or SFG probe is referred to as thepopulation period. During the population period, electronicor vibrational relaxation occurs simultaneously with orienta-tional relaxation the topic of this paper. In between any twodensity matrix elements, an arrow is drawn indicating aninteraction with an electric field. Arrows pointing toward thedensity matrix elements indicate an absorption process; ar-rows pointing away from the density matrix elements indi-cate an emission process. Absorption or emission can occuron either the bra or ket side of the density matrix element.The final signal emission is, by convention, always on theket side of the diagram. The final signal wave vector is asigned sum of the excitation field wave vectors kn, whereeach wave vector is added when it corresponds to absorptionon the ket side or emission on the bra side and subtracted

244703-3 Interfacial orientational relaxation J. Chem. Phys. 132, 244703 2010


when it corresponds to emission on the ket side or absorp-tion on the bra side. Such wave vector addition phasematching permits spatial filtering of different nonlinear sig-nals. The emission frequency is calculated in an identicalfashion; only the excitation frequencies n are summed. Thesign of the diagram is −1n where n is the number of braside interactions. The time ordering of the pulses is indicatedby the relative vertical position of their corresponding arrowson the diagram. For a more detailed explanation of the un-derlying theory of the diagrams, we refer the reader to thestandard texts.45,46

The diagrams for pump-SHG probe and pump-SFGprobe both visible and IR pump are shown in Fig. 1. Onlyphase-matched diagrams are included. Generally, the dia-grams share a similar structure in that two pump fields bringthe system into a population state. The population state is

then probed via two electric fields resulting in a coherencewhich emits the signal to form another population state. Inboth SFG and SHG experiments the probe process is theresult of both a dipole and polarizability interaction. ForSFG, the visible wave is scattered inelastically off of thevibrational coherence. For SHG, the two electric fields of thepolarizability interaction bring the system into an electroniccoherence which radiates via dipole radiation. When the non-resonant interaction is the final interaction as in SFG theemitted signal and excitation field are always on the sameside of the density matrix element.

Specifically referring to Fig. 1a, two visible electricfields interact with the bra side of the diagram, creatingpopulation in the ground state g. This population is thenexcited by two electric fields through a virtual electronicstate v to create a coherent superposition state between theground electronic state and the first excited electronic statee. This excited superposition state then oscillates at theelectronic resonance frequency emitting dipole radiation inthe phase matched direction, ksig=+2kvis. The other diagramsin Fig. 1a may be interpreted similarly. Including the signconvention discussed above, it is seen that the Feynmanpaths that proceed through ground and excited electronicpopulation states differ by a negative sign. This indicates thatthe visible-pump-SHG signal is related to the difference inelectronic polarizability upon excitation. For Fig. 1a, it hasbeen assumed that the system is a two-level electronic sys-tem. Other diagrams can be included if it is possible forhigher electronic excited states to be probed.

The diagrams for the visible-pump-SFG probe experi-ment are shown in Fig. 1b and can be interpreted in thesame manner. In this experiment, the signal not only dependson the change in the polarizability upon electronic excitationbut also any changes in vibrational oscillator strength or fre-quency induced by electronic excitation. The SFG probe stepproceeds via a dipole interaction and subsequent Raman in-teraction which is of opposite time ordering compared to theSHG probe experiment. In Fig. 1c, the diagrams for the IRpump-SFG probe experiment are shown. Here, the system isassumed to be a three level vibrational system with onlyvirtual electronic states accessible via the visible upconver-sion laser. The signal depends on ground state bleaching,stimulated emission, and excited state absorption signals.These signals are analogous to those studied by traditionalinfrared-pump-probe and two-dimensional infraredspectroscopy.34,47,48

As the underlying matter-field interactions involve non-resonant polarizability interactions, the symmetry of the mo-lecular polarizability tensor must be known in order to cal-culate the polarization dependence of the signal. To simplifyfurther calculations, the molecules being probed spectro-scopically are assumed to be symmetric tops, which permitsthe use of a diagonal polarizability tensor of the form

= 1 0 0

0 1 0

0 0 1 + − 1/3 0 0

0 − 1/3 0

0 0 2/3 , 4

where and are the isotropic and anisotropic parts of thepolarizability, respectively. This form imposes some restric-

FIG. 1. Double-sided Feynman diagrams for the a VIS-pump-SHG, bVIS-pump-SFG, and c IR-pump-SFG experiments. g, v, and e denote theelectronic ground, virtual, and excited states, respectively. 0, 1, and 2 denotethe vibrational ground and excited states, respectively. For a discussion ofthe interpretation of such diagrams, please refer to the text Sec. II A. Thetime evolution of the population state formed in the pump field-matter in-teraction is considered extensively in the text.



tion on the model, but it is a sound physical model that canbe used for a wide variety of probes at interfaces and sur-faces.

Although not denoted in the diagrams, each field canpossess its own polarization which affects the signal strengthand the orientational properties that are probed during thepopulation period. The projections of the interactions ontothe laboratory Cartesian coordinates and their spherical har-monic representations are shown in Table I. In one specialcase, the pump beam is taken to propagate along the labora-tory Z axis and circularly polarized in the XY plane.This polarization scheme was extensively applied in theliterature.19,24,25

B. The tensorial nonlinear polarization and thematerial response function

The SHGs and SFGs are nonlinear responses to multipleapplied electric fields. In the perturbative limit, the materialpolarization arising in the nonlinear processes is given by

P = 1E +

2 EE + 3 EEE

+ 4 EEEE + . . . , 5

where n is the nth-order susceptibility. The susceptibility isa tensor function of the frequency and polarization of thefields Ei denote with Greek subscripts in the general case.In a regular SHG or SFG experiment, only the second term isconsidered.10 Not every polarization combination of the in-teracting fields is allowed. At azimuthally symmetric sur-faces and interfaces Cv symmetry, only the followingsecond-order nonlinear susceptibilities are nonzero:ZZZ

2 ,ZXX2 =ZYY

2 ,XZX2 =YZY

2 ,XXZ2 =YYZ

2 , where the polar-izations are given in the laboratory frame. The material po-larization in TRSHG probe and TRSFG probe experiments isequal to the fourth term of the perturbative expansion in Eq.5. The pump field polarization can be any linear combina-tion of the X, Y, and Z coordinates. A circularly polarizedpump beam can also be applied that usually propagates along

the Z axis.19,24,25 Although the dynamics of the molecularsystem is contained in n in the frequency domain, theknowledge of the time-domain material response functionRn is more practical for time-resolved experiments. Thetime-domain material response function is the inverseFourier-transform of the nonlinear susceptibilities46,45

R. . .n tn, . . . ,t1 =

1

2n−

dn. . .−

d1. . .n

exp− i1 + . . . + ntn . . .

exp− i1t1 . 6

In Eq. 6, Rn is the n+1-time correlation function thatdescribes the system for a given time ordering. It is a tensorfunction that retains the symmetry properties of the nonlinearsusceptibility n. Generally, it is a valid assumption that theorientational motions are decoupled from the vibronic modesof the system, and the time-domain material response func-tion can be separated into contributions from the vibrationaland orientational degrees of freedom,

R. . .n tn, . . . ,t1 = R. . .

n tn, . . . ,t1RVtn, . . . ,t1 , 7

where R. . .n tn , . . . , t1 is the orientational response function

and RVtn , . . . , t1 is the vibrational response function.49 Onlythe effect of R. . .

n tn , . . . , t1 on the nonlinear material polar-ization will be discussed here.

The pump interactions are taken to be instantaneous. Thetime-dependences of the emission coherence for the SHGprobe case or the vibrational coherence for the SFG case donot influence the orientational relaxation observables duringthe population period, and will not be considered. Withinthese conditions, the orientational correlation function can begreatly simplified and expressed as a two-time correlationfunction,

TABLE I. Resonant and nonresonant interactions projected onto the laboratory frame and expressed withspherical harmonics. For the nonresonant interactions, a diagonal polarizability tensor is assumed.=+2 /3, =− /3 see Eq. 4.

Interaction Projection onto the laboratory frame Spherical harmonic representation

Resonant interactionsiZ cos i 2 /31/2Y1

0iiY sin i sin i i2 /31/2Y1

−1i+Y11i

iX sin i cos i 2 /31/2Y1−1i−Y1

1iiCIRC sin i 1− 4 /3Y1

0i21/2

Nonresonant interactionsZZ cos2 i+ sin2 i + 4 /3 /5Y2

0iYY + −sin2 i sin2 i −2 /3 /51/2Y2

0i+ 2 /151/2Y2−2i+Y2

2iXX + −sin2 i cos2 i +−2 /3 /51/2Y2

0i+ 2 /151/2Y2−2i+Y2

2iYY −cos i sin i sin i i2 /151/2Y2

−1i+Y21i

ZYXZ −cos i sin i cos i 2 /151/2Y2

−1i−Y21i

ZXXY −sin i sin i cos i i2 /151/2Y2

−2i−Y22i

YX



RSHG t1 = d1 d011G1,t1002,

8

RSFG t1 = d1 d011G1,t1002,

9

where G1 , t 0 is a time propagator or Green’s functionsee below. A shorthand notation can be introduced to writethe above equations in an abbreviated form as

RSHG t1 = 10 10

and

RSFG t1 = 10 . 11

The simultaneity of the final three interactions imposes ad-ditional symmetry conditions on the correlation functionsand permits the interchange of certain pulse pairs, similar tothe Kleinman symmetry see Sec. III A.10

The physical interpretation of the above equations is in-tuitive: after the electronic or vibrational excitation with thepump pulse, the population state see Fig. 1 evolves in time.The time evolution is given by the G1 , t 0 operator,which is the joint probability of finding the probe at orienta-tion 1 at time t1, provided it was at orientation 0 at t=0.Then, at t1, the state of orientation is projected onto the probeand signal polarizations. In the general description, the time-evolution operator the Green’s function depends on themodel of reorientation. In the free rotation, for example, itcan be expanded as a series of spherical harmonics32,50

Gi,ti0 = n=0

m=−n

n

cnmtiYn

m

0Ynmi , 12

where the expansion coefficients are given by

cnmti = exp− nn + 1Dorti 13

and the spherical harmonics are defined over the angular co-ordinates in the laboratory frame see below. For re-stricted rotational diffusion within a particular solid angle orcone, the time-evolution operator has a different form, andthe integration is taken only over the restricted angular vol-ume.

C. Definition of angular variables

As mentioned above, the molecules will be treated asrigid, cylindrically symmetric rods although the formalismcan be extended to other shapes. For the nonresonant inter-actions, the molecules are taken to be symmetric tops, whichis consistent with a cylindrical shape. A symmetric top sim-plifies the results because the polarizability tensor is diago-nal. The IR and electronic transition dipoles lie along thesymmetry axis of the polarizability of the rod. The procedurecan be generalized to any other polarizability tensor symme-try.

The measured correlation functions will be calculated asa two-time correlation function. If the orientation of the mol-

ecule is given in the polar coordinates by = ,, and theinteractions are represented with spherical harmonics, the po-larization dependent components of the tensorial correlationfunction will be obtained in the form of dynamic ensembleaverages of spherical harmonics: Yl

mtYlm0. After

obtaining the correlation functions in this general form, thewobbling-in-a-cone model will be applied to express them asexponential decays.

Figure 2 illustrates the relevant angular variables for thesurface experiments. The excitation and output beams propa-gate in the XZ plane. Their incident angle is measured fromthe Z axis and denoted by i. The beams are polarized fromthe vertical with the angle i. This gives the projection of theelectric field onto the Cartesian axes as

i = icos i cos iZ + cos iY + cos i cos iX .

14

The orientation of a molecule in the laboratory frame at anytime can be given by the polar and the azimuthal anglesLMt= LMt ,LMt.

It is practical to define a coordinate system that is fixedto the conical volume in which the orientational diffusion ofthe molecules is restricted. The Z axis of the conical frame isthe symmetry axis of the cone, and if it is tilted from thelaboratory Z axis by the angle LC, the Euler angles thattransform the laboratory frame into the conical frame areLC,LC,LC. The Euler angles are not functions of thetime and can be factored out from the laboratory frame cor-relation functions.

Within the conical frame, the orientation of the moleculeis given by the polar and the azimuthal angles CMt

εiΩ

iϕ

LMφ

LMθ

μ

ZL

XL

YL

Ei

CMθ

ZC

XC

YC

μ

LMφ

0θ

FIG. 2. Angular variables in the laboratory frame upper and the conicalframe lower. The sample surface is in the XY plane. The probe and signalbeams propagate in the XZ plane incident from the ZL axis by the angle i

and polarized from the vertical by the angle i. o denotes the cone half-angle and the orientation of a molecule is given by the polar and azimuthalangles LMt= LMt ,LMt and CMt= CMt ,CMt in the labora-tory and conical frames, respectively.



= CMt ,CMt. In the cone model, the polar angle is re-stricted to 0CMto, where o denotes the half-coneangle.

Using SFG or SHG, it is possible to measure the aver-age angle of orientation of a molecule. Within the tilted conemodel the average angle can be expressed as

= LC−o

LC+o sin d

LC−o

LC+osin d, 15

which is equivalent to

cos =1 + cos o

2, 16

when LC=0.18,51

D. Orientational correlation functions for restrictedrotational diffusion

As mentioned above, the orientational relaxation of themolecules at a surface or interface cannot be described asisotropic diffusion as is the case in bulk liquids. Instead, weadopt the model of wobbling-in-a-cone that has proved to beuseful in a numerous of previous problems.31,42,52–55 To sim-plify the problem, molecules are assumed to be rigid rodsthat diffuse freely within a given angular volume, the cone.This model was also suggested by Shen and co-workers51 forinterfacial OH bonds. The orientation of a molecule at agiven time can be specified by its polar coordinates withinthe cone CMt= CMt ,CMt.

The diffusion in a cone model was first solved by War-chol and Vaughan.29 Later Wang and Pecora obtained theYl

mtYlm0 l=1,2 and m=0,1 ,2 correlation func-

tions within this model.30

With conical boundary conditions and with initial condi-tion

c,0 = − 0 = cos − cos 0 − 0 ,

17

the general solution of the rotational diffusion equation is

c,t = n=1

m=−

exp− nmn

m + 1DortYnm

m

0Yn

mm t ,

18

where c , t is the probability density for finding the rod inorientation at time t, Dor is the diffusion constant Y

nm

m

denotes the associated spherical harmonics, nm depends on

the cone half-angle and can be fractional unlike in the freediffusion model, and the indicates the complex conjugateof the associated spherical harmonic.56,57

The joint probability of finding a rod with orientation0 in solid angle d0 at time t=0 and orientation tin solid angle dt at time t is then,

Gst,t0,0

=1

21 − cos on=1

m=−

exp− nmn

m + 1Dort

Yn

mm

0Yn

mm t , 19

where the Green’s function satisfies the normalization condi-tion

Gst,t0,0dtd0 = 1. 20

The angular integrals are taken only over the conical volumeaccessible to the molecule through diffusion.

These results lead to the orientational correlation func-tions that can be written as ensemble averages of sphericalharmonics

Ylm

tYlm0

= d0 dtYlm

tGst,t0Ylm0 .

21

In general, the ensemble averages have the following form:

Ylm

tYlm0

=mm

42l + 12l + 1l − m!l − m!

l + m!l + m!

n=1

Cnm exp− n

mnm + 1Dort 22

and

Ylm

tYlm0 = Yl

−m

tYl−m0 , 23

i.e., the orientational correlation function is an infinite sumof exponential decays.

The Cnm coefficients are calculated as

Cnm =

1

Hnm1 − cos o

0

o

d sin Plmcos P

nm

m cos

0

o

d sin Plmcos P

nm

m cos , 24

where

Hnm =

0

o

d sin Pn

mm cos P

nm

m cos . 25

Using the integrals from Eqs. 24 and 25 in Eq. 22,the orientational correlation functions for any l and m can becalculated. Unfortunately, most of these integrals can only becalculated numerically. However, for each l and m, it isshown in Appendix A that the summation can be truncatedafter the first decaying exponential term because the combi-nation of all of the other terms is negligible for cone half-angles that will be encountered in the vast majority of sys-tems. In addition, Appendix A also shows that for the SHG



and SFG experimental observables, the necessary coeffi-cients Cn

m can be obtained analytically in terms of the conehalf-angle. Furthermore, while the n

m coefficients can onlybe calculated numerically, they can be well approximated byanalytical formulae for o170°, which is a half-angle thatis larger than expected for most surface and interfacial sys-tems. The net result is that Eq. 22 reduces to a tractableexpression that can be accurately calculated analytically. Thenecessary terms are either a single exponential or a constantplus a single exponential.

We will also need to calculate both Ylm

t andYl

m0. The average value of these single spherical har-monics is

Ylm

t = Ylm0

=m0

1 − cos o2l + 1

4

0

o

Plcos sin d ,

26

where Pl is the lth order Legendre polynomial.33,58

For the isotropic case, Wang and Pecora30 calculated the

Ylm

tYlm0 l=1,2 correlation functions. How-

ever, for TRSHG and TRSFG experiments, different correla-tion functions are needed and are derived below in a mannersimilar to the l=1,2 cases. In the end, the necessary corre-lation functions are obtained as exponential decays.

Armed with expressions for the time correlation func-tions of spherical harmonics, it will now be possible to cal-culate the exact orientational response given the experimen-tal polarization conditions for a TRSHG or TRSFGexperiment. In Sec. III, we will first demonstrate the properexpansion of the orientational response into spherical har-monics and then plug in the results from the above section.

III. RESULTS FOR TRSHG AND TRSFGEXPERIMENTS

A. The orientational correlation functions in thelaboratory frame

First the orientational correlation functions RSHG and

RSFG will be calculated in the laboratory frame. It requires

the evaluation of the integrals in Eqs. 8 and 9. The time-evolution operator G1 , t 0 is not evaluated in this stepand the orientational correlation functions will be obtained ina general form as a sum of the dynamic ensemble averages

of spherical harmonics Ylm

tYlm0.

The electric-field-dipole and electric-field-polarizabilityinteractions are transformed into the laboratory frame andrepresented with spherical harmonics in Table I. When thefields are temporally and spatially overlapped, the sphericalharmonic representations of the interactions can always bewritten as the sums of a constant and spherical harmonics ofthe order from one to three. The purely algebraic transforma-tions of the spherical harmonic representations are given indetail in Appendix B for completeness.

Symmetry relations can reduce the number of correlationfunctions that need to be calculated. The simultaneity of thefinal three interactions see Eq. 4, Table I, and Fig. 1 per-

mits the interchange of certain pulse pairs, and one can ob-tain the following symmetry relationships the electric fieldis either linearly polarized along the Cartesian axes X, Y, andZ or circularly polarized in the XY plane denoted by C in thesubscript:

RZZZZZSHG = RZZZZZ

SFG ,RZZZYYSHG = RZZZXX

SHG = RZZZYYSFG = RZZZXX

SFG , 27

RZYYZZSHG = RYYZZZ

SFG ,RZYYYYSHG = RYYZYY

SFG ,RZYYXXSHG = RYYZXX

SFG , 28

RZXXZZSHG = RXXZZZ

SFG ,RZXXYYSHG = RXXZYY

SFG ,RZXXXXSHG = RXXZXX

SFG , 29

RYYZZZSHG = RYZYZZ

SHG = RYZYZZSFG = RZYYZZ

SFG ,RYYZYYSHG = RYZYYY

SHG

= RYZYYYSFG = RZYYYY

SFG ,

30RYYZXX

SHG = RYZYXXSHG = RYZYXX

SFG = RZYYXXSFG ,

RXXZZZSHG = RXZXZZ

SHG = RXZXZZSFG = RZXXZZ

SFG ,RXXZYYSHG = RXZXYY

SHG

= RXZXYYSFG = RZXXYY

SFG

31RXXZXX

SHG = RXZXXXSHG = RXZXXX

SFG = RZXXXXSFG ,

RZZZCCSHG = RZZZCC

SFG ,RZYYCCSHG = RYYZCC

SFG ,RZXXCCSHG = RXXZCC

SFG , 32

RYYZCCSHG = RYZYCC

SHG = RYZYCCSFG = RZYYCC

SFG ,RXXZCCSHG = RXZXCC

SHG

= RXZXCCSFG = RZXXCC

SFG . 33

It is sufficient to calculate the orientational correlation func-tions with the SHG time ordering. The correlation functionswith the SFG time ordering can be obtained by using theabove symmetry relations. At the azimuthally symmetricCv symmetry interfaces, the number of the unique orien-tational correlation functions is further reduced

RZZZYYSHG = RZZZXX

SHG , 34

RZYYZZSHG = RZXXZZ

SHG ,RZYYYYSHG = RZXXXX

SHG ,RZYYZZSHG

= RZXXZZSHG ,RZYYCC

SHG = RZXXCCSHG , 35

RZYYYYSHG = RZXXXX

SHG ,RZYYXXSHG = RZXXYY

SHG , 36

RYYZZZSHG = RYZYZZ

SHG = RXZXZZSHG = RXXZZZ

SHG ,RYYZCCSHG = RYZYCC

SHG

= RXZXCCSHG = RXXZCC

SHG , 37

RYYZYYSHG = RYZYYY

SHG = RXZXXXSHG = RXXZXX

SHG ,RYYZXXSHG = RYZYXX

SHG

= RXZXYYSHG = RXXZYY

SHG . 38

Therefore, the unique orientational correlation functions thatneed to be calculated are

RZZZZZSHG , RZZZYY

SHG , RZYYZZSHG , RZYYYY

SHG , RZYYXXSHG , RYYZZZ

SHG ,

RYYZYYSHG , RYYZXX

SHG , RZZZCCSHG , RZYYCC

SHG ,

and

RYYZCCSHG .



Now, taking the spherical harmonic representations ofthe field-matter interactions listed in Table I, the orientationalcorrelation functions can be calculated. The detailed deriva-tion is given here for the RZZZZZ

SHG tensor element only. For theother tensor elements, the derivations are similar and theresults are listed in Table II.

Using the shorthand notation introduced in Eqs. 10 and11,

RZZZZZSHG t1 = ZZZ1ZZ0 . 39

Substituting the spherical harmonic representations of the in-teractions from Table I and Appendix B, one can obtain

RZZZZZSHG t1 = 2

15

315 + 4Y1

0t1

+4

5

7Y3

0t11

3+

4

3

5Y2

00 ,

40

which yields

2

472515 + 42815Y1

0Y20 + 353Y1

0

+ 9785Y30Y2

0 + 10Y30 , 41

where the following notation is used:

YlmY2

m = Ylm

labtGSlabt,t;lab0,0Y2mlab0 .

42

The averaging is over the polar and azimuthal angles at timet1 and t=0.

Without applying any orientational diffusion model, thecorrelation functions are listed in Table II. One can see thatthe C1t and C2t correlation functions do not appear in theequations. Instead, the

YlmY2

m

= d0 dtYlm

tGst,t0Y2m0

l = 1,3; m = 0,2 43

correlation functions are measured in the TRSHG andTRSFG experiments. The integrals are performed only overthe volume in which the orientational diffusion occurs. Thetime-evolution operator Gst , t 0 depends on themodel of the orientation relaxation and the potential withinthe volume of diffusion. In certain cases, the operator can berepresented by spherical harmonics, which permits the deri-vation of analytical forms of the correlation functions. If thetime-evolution operator cannot be represented in such a way,the correlation functions can be evaluated numerically.

TABLE II. The orientational correlation functions for tensor elements of the fourth-order polarization for theTRSHG experiments calculated from Eq. 8. For simplicity, the following notation is used: Yl

mY2m

= Ylm

labtGSlabt , t ;lab0 ,0Y2mlab0. and are the isotropic and anisotropic part of the polar-

izability tensor see Eq. 4.

RZZZCC4

472515 + 4− 1415Y1

0Y20 + 353Y1

0 − 9745Y30Y2

0 − 10Y30

RZYYCC4

472515 − 2− 1415Y1

0Y20 + 353Y1

0 + 9725Y30Y2

0 − 5Y30

RYYZCC4

1575− 1415Y1

0Y20 + 353Y1

0 + 635Y30Y2

0 − 157Y30

RZZZZZ2

472515 + 42815Y1

0Y20 + 353Y1

0 + 9785Y30Y2

0 + 10Y30

RZZZYY2

472515 + 4− 1415Y1

0Y20 + 353Y1

0 − 9745Y30Y2

0 − 10Y30

RZYYZZ2

472515 − 22815Y1

0Y20 + 353Y1

0 − 9745Y30Y2

0 + 5Y30

RZYYYY

2

472515 − 2− 1415Y1

0Y20 + 353Y1

0 + 9725Y30Y2

0 + 10Y32Y2

2

− 5Y30

RZYYXX2

472515 − 2− 1415Y1

0Y20 + 353Y1

0 + 9725Y30Y2

0 − 10Y32Y2

2 − 5Y30

RYYZZZ2

15752815Y1

0Y20 + 353Y1

0 − 1235Y30Y2

0 − 157Y30

RYYZYY2

1575− 1415Y1

0Y20 + 353Y1

0 + 635Y30Y2

0 + 307Y32Y2

2 − 157Y30

RYYZXX

2

1575− 1415Y1

0Y20 + 353Y1

0 + 635Y30Y2

0 − 307Y32Y2

2 − 157Y30



B. The ŠYlmY2

m‹ correlation functions in a tilted

restricted cone

In Sec. III A, the spherical harmonic correlation func-tions for a restricted cone model were found. However, toproperly represent any molecular orientation at the interface,it is necessary to allow for the cone to be tilted with respectto the surface normal. Here we incorporate this “tilted” frameof reference into the model.

In an empty cone whose symmetry axis is the surfacenormal, the polar coordinates of the molecules within thelaboratory LM are equal to the coordinates in the coneCM frame, i.e., LMt=CMt. If the symmetry axis ofthe cone is tilted from the surface normal, the correlationfunction written in the laboratory frame has to be trans-formed into the conical frame. This case corresponds to amolecule which reorients within the polar angle intervala ,b.

According to the addition theorem of sphericalharmonics,58

YlmLMt,LMt

= m

Dmml LC,LC,LCYl

mCMt,CMt ,

44

where LM,LM are the polar coordinates of the moleculein the laboratory frame, LC,LC,LC are the Euler anglesthat transform the laboratory frame into the conical frame,and CM,CM are the polar coordinates of the molecule inthe conical frame. Dmm

l denotes the lth order Wigner D ma-trix.

The Euler angles that connect the laboratory frame withthe conical frame are constant, while the polar angles in theconical frame evolve in time. Because the interface and thecone are taken to have a Cv symmetry, LC=0. Every LC

angle is equally probable; therefore, one can average over theazimuthal angles, and only the tilt angle from the surfacenormal matters.

Using the orthogonality properties of the ensemble aver-ages of spherical harmonics

Ylm = m0Yl

0 45

and

YlmYl

m = mmYlmYl

m , 46

the correlation functions derived in the laboratory frame canbe transformed into the conical frame. The laboratory framecorrelation functions expressed in the conical frame are listedin Table III.

To illustrate how to transform the correlation functionsfrom the laboratory frame to the conical frame, we examinethe RZZZZZ

SHG tensor element derived above. According to Eq.41, it consists of the Yl

0Y20 l=1,3 correlation functions

and the Yl0 ensemble averages, which can be transformed

into the tilted frame individually by using the equationslisted in Table III. The Wigner small d matrices can be foundin handbooks58 and the correlation functions in the rotatedframe are the following:

Y10Y2

0L = 3Y11Y2

1Csin2 LC cos LC

+ 12 Y1

0Y20Ccos LC3 cos2 LC − 1 , 47

Y30Y2

0L =35

4Y3

2Y22Ccos LC sin4 LC +

3

42Y3

1Y21C

3 + 5 cos2LCcos LC sin2 LC

+1

16Y3

0Y20C3 cos LC + 5 cos3LC

3 cos2 LC − 1 , 48

where the subscript C denotes an ensemble average in theconical frame.

It can be seen that when LC=0, the YlmY2

m m=1,2correlation functions vanish in the sums and the correlationfunctions expressed in the laboratory frame and in the coni-cal frame are identical.

The ensemble averages are obtained as

Y10L = Y1

0Ccos LC, 49

and

Y30L = 1

8 Y30C3 cos LC + 5 cos3LC . 50

Substituting these expressions into Eq. 41, one can getthe RZZZZZ

SHG tensor element for the tilted cone case as

TABLE III. Rotating correlation functions into the tilted cone frame. The symmetry axis of the cone is tiltedfrom the surface normal by the angle LC, which does not evolve in time. The relationship between thecorrelation functions in different frames are given by the addition theorem of spherical harmonics in Eq. 44.dj,k

i denotes the small Wigner d-matrix see Sec. IV B.

Y10Y2

0 = j=−1

1

Y1jCMt,CMtY2

j CM0,CM0d0j1 LC− 10−jd0−j

2 LC

Y30Y2

0 = j=−2

2

Y3jCMt,CMtY2

j CM0,CM0d0j3 LC− 10−jd0−j

2 LC

Y32Y2

2 = j=−2

2

Y3jCMt,CMtY2

j CM0,CM0d2j3 LC− 12−jd−2−j

2 LC

Y10= Y1

0CMt ,CMtd001 LC

Y30= Y3

0CMt ,CMtd003 LC



RZZZZZSHG =

2

67515 + 4cos LC5Y1

0Y20C

6 + 3 + 33 − 2cos2LC +45

47Y3

0C

3 cos LC + 5 cos3LC +9

275Y3

0Y20C

3 cos2LC − 13 cos LC + 5 cos3LC

+9

276 cos LC sin2 LC10Y1

1Y21C

3 + 5 cos2LC + 10Y32Y2

2Csin2 LC . 51

C. The correlation functions as exponential decays

Within the conical frame, the dynamic ensemble aver-ages Yl

mY2m l=1,3 , m=0,1 ,2 have to be calculated to

describe the orientational relaxation of the molecules. Wangand Pecora30 solved the Yl

mYlm l=1,2 , m=0,1 ,2 orien-

tational correlation functions for restricted diffusion and wefollow their procedure to obtain the correlation functions todescribe the TRSHG and TRSFG experiments. The details ofthe mathematics are given in Appendix A and here we showonly the resulting equations. We used the model of restricteddiffusion in a conical volume as outlined in the Sec. II D seeabove. According to Eq. 23, it is sufficient to solve theproblem for positive m values only, and the correlation func-tions can be written as an infinite sum of exponential decays.The coefficients of the exponentials can be calculated nu-merically and it is shown in Appendix B that the sums canalways be truncated after the first decaying exponential term.

Table IV shows the results. It can be seen that the cor-relation functions can be written in the following generalform:

Yl0

tY200 = Ao + Boexp− 2

020 + 1Dort,

l = 1,3 52

and

Ylm

tY2m0 = Aoexp− 1

m1m + 1Dort,

l = 1,3 and m 0 . 53

It is important to point out that the 20 and 1

m constants de-pend on the cone half-angle o. While they can only be cal-culated exactly numerically, for a very wide range of conehalf-angles o170°, they can be very well approximated bythe following formulae:31,38,56,57

20 = 1

2 = 100.496o−1.122 54

and

11 = 100.237o

−1.122, 55

where o is in radian. The 11 coefficient appears only in the

tilted cone cases. When the tilt of the cone is not significant,the correlation functions in the conical frame are given as aconstant plus a single exponential decay that depend on thehalf-cone angle. The offset in the tilted case is also nonzeroand depends on the ensemble averages of certain sphericalharmonics see Table III.

At this point, we have derived the correlation functionsfor the restricted diffusion in a cone and the nonzero orien-tational correlation functions in the laboratory frame. Usingthe optical beam geometry, the observed pump-SHG probeand pump-SFG probe orientational correlation functions canbe constructed from the equations that were derived.

To summarize, to calculate the time correlation func-tions, one projects the excitation and signal beams onto Car-tesian coordinates in the laboratory frame Table V and thelaboratory frame correlation functions onto the tilted conicalframe Table III. Then, the polarization selective orienta-tional correlation functions in the laboratory frame can be

TABLE IV. Correlation functions in the conical frame. For transforming the correlation functions from theconical frame into the laboratory frame, see Table III.

Y10Y2

0 =15

41

4cos o1 + cos o2 +

1

4sin2o

2sin2 oe

−202

0+1DortY1

1Y21 =

35

32

sin4o1 − coso

e−111

1+1Dort

Y30Y2

0 =35

4 1

16cos o1 + cos o25 cos2 o − 1 +

1

4cos2o

2sin4o

29 + 10 cos o + 5 cos 2oe−2

020+1Dort

Y31Y2

1 =35

6427 + 5 cos2osin4o

1 − cosoe−1

111+1Dort

Y32Y2

2 =57 sin6 o

641 − cos oe−1

212+1Dort

Y10 =

1

4 3

1 + cos o

Y30 =

1

16 7

cos2o

23 + 5 cos2o



calculated according to Table II. In Sec. IV, calculations arepresented to demonstrate the dependence of the signal on theorientational diffusion constant and the cone half-angle.

IV. MODEL CALCULATIONS AND ANALYSIS OFEXPERIMENTS

A. Dependence of the observables on the conehalf-angle

The effects of varying the diffusion constant and thecone half-angle will be demonstrated by calculating the ori-entational correlation function for a particular polarizationcombination, that is, the tensor elements for the circularlypolarized pump scheme. Here, for simplicity, we will takethe excited state life time to be long compared to the orien-tational relaxation, and leave it out of the calculations. Be-cause the orientational motion is decoupled from the vi-bronic degrees of freedom see Eq. 7, the effect of thevibrational or electronic relaxation can be included by mul-tiplying by the exponential population decay

RL t = exp− kRtRt , 56

where kR is the excited state relaxation rate and the super-script L indicates that the population decay is included. Anexcited state lifetime much longer than the orientational re-laxation time scale commonly occurs in a TRSHGexperiment.25

The simplest experimental scheme has the pump beamincident normal to the sample surface and circularly polar-ized in the plane of the surface.19,25 The pump pulse elec-tronically excites the molecules through an allowed one-photon absorption. The conical volume of the diffusion isassumed to be a cone whose symmetry axis is aligned alongthe surface normal. The lifetime decay is not included. Whenthis approximation is invalid, it is necessary to measure de-cays at multiple pump-probe polarization conditions to sepa-

rate the orientational decay from the population decay. ForTRSHG and TRSFG experiments there is no analog to theanisotropy rt Eq. 1.

To perform the model calculations, we use the equationsthat were derived for the orientational correlation functionsand are listed in Table II. For the calculations, the inputparameters are the cone half-angle, the orientational diffu-sion constant, and the molecular polarizability. All the calcu-lations are for the case when the probe beam is two-photonresonant with an electronic excited state of the molecule. Forexample, the pump pulse is at 400 nm and the probe pulse isat 800. The electronic transition is both one-photon and two-photon resonant.

Figure 3 shows the simulated RZYYCC tensor componentof the orientational correlation function. The diffusion con-stant is set to a value of 10−3 ps−1, which is a realistic value

TABLE V. Polarization dependence of the TRSHG and TRSFG experiments. The beams are incident on thesample surface at the angle from normal and polarized at from the vertical

P4 = RZZZZZ cos SHG cos2 VIS cos2 pump sin SHG sin2 VIS sin2 pump

+RZZZYY cos SHG cos2 VIS sin2 pump sin SHG sin2 VIS

+RZZZXX cos SHG cos2 VIS cos2 pump sin SHG sin 2 VIS cos 2pump

+RZYYZZ cos SHG sin2 VIS cos2 pump sin SHG sin2 pump

+RZYYYY cos SHG sin2 VIS sin2 pump sin SHG

+RZYYXX cos SHG sin2 VIS cos2 pump sin SHG cos2 pump

+RZXXZZ cos SHG cos2 VIS cos2 pump sin SHG cos2 VIS sin2 pump

+RZXXYY cos SHG cos2 VIS sin2 pump sin SHG cos2 VIS

+RZXXXX cos SHG cos2 VIS cos2 pump sin SHG cos2 VIS cos2 pump

+RYZYZZ sin SHG cos VIS sin VIS cos2 pump sin VIS sin2 pump

+RYZYYY sin SHG cos VIS sin VIS sin2 pump sin VIS

+RYZYXX sin SHG cos VIS sin VIS cos2 pump sin VIS cos2 pump

+RXZXZZ cos SHG cos2 VIS cos2 pump cos SHG sin VIS cos VIS sin2 pump

+RXZXYY cos SHG cos2 VIS sin2 pump cos SHG sin VIS cos VIS

+RXZXXX cos SHG cos2 VIS cos2 pump cos SHG sin VIS cos VIS cos2 pump

+RYYZZZ sin SHG sin VIS cos VIS cos2 pump sin VIS sin2 pump

+RYYZYY sin SHG sin VIS cos VIS sin2 pump sin VIS

+RYYZXX sin SHG sin VIS cos VIS cos2 pump sin VIS cos2 pump

+RXXZZZ cos SHG cos VIS cos VIS cos2 pump sin SHG cos VIS sin VIS sin2 pump

+RXXZYY cos SHG cos VIS cos VIS sin2 pump sin SHG cos VIS sin VIS

+RXXZXX cos SHG cos VIS cos VIS cos2 pump sin SHG cos VIS sin VIS cos2 pump

0 100 200 300 400 500 6000.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

R yyzcc(θo,Dor)(arb.units)

t (ps)

τ = 147 ps θo = 80°

τ = 85 ps θo = 60°

τ = 48 ps θo = 45°

τ = 21 ps θo = 30°

Dor = 10-3 ps-1

FIG. 3. The RYYZCCSHG tensor element is calculated for different cone semi-

angles and shown as a decay. The bleach in the visible-pump-SHG experi-ment is shown. The diffusion constant is set to an arbitrary but realisticvalue D=10−3 ps−1, while the cone semiangle is varied between 30° and80°. The SHG/SFG signal always recovers exponentially to an equilibriumvalue. The exponential time constant of each decay is indicated in the figure;it is clear that the apparent time constant and the offset both depend stronglyon the cone half-angle.



for molecules like Rhodamine 6G Ref. 19 or C314 Ref.27, for example. The isotropic part of the polarizability isset to 1, and the anisotropic part is set to 1/3, that is, theanisotropic-isotropic ratio is one third. The cone semiangle isvaried between 30° and 80°. Before t=0, the SHG signal ofthe sample is recorded. When the pump beam is turned on,the SHG signal decreases, and it recovers to the equilibriumvalue over time. As an example, the RYYZCC

SHG response func-tion is plotted as a function of the pump-probe delay time inFig. 3. Each of the signal decay curves is a single exponen-tial that decays to an offset. Both the decay constant and theoffset depend both on the orientational diffusion constant andthe cone half-angle. In Fig. 3, all of the curves are calculatedfor the same diffusion constant, Dor=10−3 ps−1. The conehalf-angles are varied as shown in the figure. To obtain a feelfor how much the cone half-angle influences the experimen-tally observed decay time, the curves were fit to exponentialdecays to an offset. The results are dramatic. Although Dor isthe same for all curves, the apparent decay time constantsrange from 21 o=30° to 147 ps o=80°. It is clear fromthese model calculations that the decay constant alone cannotprovide microscopic information on the orientational dynam-ics. However, as discussed in Sec. IV B, if the cone half-angle is known from time-independent measurements,11 thenmeasurement of the decay does permit the determination ofthe orientational diffusion constant.

The theory shows that it is not necessary to make a sepa-rate static measurement of the cone half-angle. It is possibleto obtain both the orientational diffusion constant and thecone half-angle from the time-dependent experiment alone.In Fig. 3, the initial amplitude of the signal varies with thecone half-angle. In addition, the ratio of the initial valuet=0 and the long time plateau amplitude is sufficient todetermine the cone half-angle. With the cone half-angle andthe decay constant, the orientational diffusion constant canbe determined.

Figure 4 shows three decay curves for the same conehalf-angle 45°, but with diffusion constants varying by twoorders of magnitude 10−2–10−4 ps−1. The figure shows sev-eral important points. First, for the same cone half-angle, theinitial value of the signal is the same independent of the

diffusion constant. Furthermore, the plateau level is the samefor all three curves, although the slowest decay in the figurehas not reached this plateau in the time range displayed. Asdiscussed in connection with Fig. 3, the initial value and theplateau value are sufficient to determine the cone half-angle.With the cone half-angle and the decay time constant, theorientational diffusion constant Dor can be determined.

If the decays in Fig. 4 are fit to an exponential decayingto a constant, and the decay time constant is taken to be =1 /6D, as it would be in a bulk liquid, the diffusion con-stants are always a factor of 3.5 larger than the diffusionconstant used to calculate the decays. For other cone half-angles, the factor is different as is clear from Fig. 3. As withFig. 3, Fig. 4 shows that it is not possible to determine theorientational diffusion constant from a measurement of thedecay time. Furthermore, Figs. 3 and 4 demonstrate that de-cay times measured on different systems, which can havedifferent cone half-angles, cannot be directly compared.

B. A realistic example: TRSHG measurementson the C314 monolayer

Above, calculations were presented with several diffu-sion constants and half-cone angles to illustrate how the ob-servables depend on both of these physical parameters. Herewe examine a system that has been examined experimentally.Eisenthal and co-workers studied C314 at the air/water inter-face of an aqueous solution by TRSHG24,25 and TRSFG Ref.22 experiments as well as in bulk water.59 In bulk water, theorientational relaxation time constant was found to be 106ps.60 The surface relaxation time constant was also measuredat the air/water interface, and it was found to be 350 ps.24

From the comparison of the measured time constants, theauthors concluded that the orientational diffusion is signifi-cantly slower at the interface.

Nguyen et al.25 modified the surface by adding the sur-factant sodium dodecyl sulfate SDS to the C314 solution.SDS forms a monolayer at the air-water interface. At thesurface, the C314 orientational time constant was found to bevery similar to the interface without the surfactant, only in-creasing 15% to 383 ps. From the time constants, it wasconcluded that modifying the surface changes the C314 ori-entational diffusion only slightly.

By using Eq. 3, the diffusion constant in the bulk phasecan be obtained, and it is 1.6 10−3 ps−1. To obtain the dif-fusion constants at the interfaces, we use the equations de-rived above. In the analysis, we employ the previously pub-lished experimental data24,25 to demonstrate how theorientational correlation functions along with the wobbling-in-a-cone model can be used to obtain the orientational dif-fusion constants at the air-water interface and the air-water-surfactant interface. Eisenthal and co-workers25 measuredthe SHG signal using the XZX

2 and ZXX2 susceptibility tensor

elements, and found identical reorientational dynamics forboth tensor elements. They report that after excitation, theSHG signal decreases and it recovers exponentially to anequilibrium value. The time-dependent signal can be fit witha single exponential decay plus a constant. At the air-water

0 200 400 600 800 10000.0090

0.0095

0.0100

0.0105

0.0110

0.0115

0.0120

Dor = 10-3 ps-1

R yyzcc(θo,Dor)(arb.units)

θo = 45°

t (ps)

Dor = 10-2 ps-1

Dor = 10-4 ps-1

FIG. 4. The RYYZCCSHG tensor element calculated for different diffusion con-

stants. The bleach in the visible-pump-SHG experiment is shown. The conehalf-angle is set to o=45°, while the diffusion constant is varied10−4–10−2 ps−1. The SHG/SFG signal always recovers to the same equilib-rium value independently of the diffusion constant. If it is assumed that =1 /6Dor, the obtained diffusion constant is larger by a factor of 3.5 fromwhat was used in the simulations.



interface, the time constant of the single exponential decay is350 ps, and the remaining bleach is approximately 20%. Theexact number of the offset was not reported.

To use the equations derived above, one needs the conehalf-angle through which restricted diffusion occurs. In prin-ciple, the cone half-angle can be obtained from the offset ofthe exponential decay see Figs. 3 and 4. In this case, onlyan approximate value of the offset was reported. The conehalf-angle can also be obtained from the average orienta-tional angle which was determined in static experiments.25 InEqs. 15 and 16, we show that the average orientationalangle of the molecules at the surface can be related to thecone half-angle by an orientational average. At the air-waterinterface, the measured average polar angle of the C314 mol-ecules is 67°,25 which corresponds to a cone half-angle of102°, a reasonable value. In molecular dynamics studies, itwas found that at the air-water interface the C314 moleculescan lie in the plane of the surface and also that the less polarend of the molecule can rotate to penetrate into the bulkphase.61,62

From the equations in Table II, one can see that thecombination of the Y1

0Y20 and Y3

0Y20 correlation functions

are measured by the experiment. We assume that the symme-try axis of the cone of the diffusion is the surface normal andthe dipole moment of C314 lies in the molecular plane.Therefore, the correlation functions decay as single exponen-tials in every combination as observed. The decay rate is2

020+1Dor. The coefficients in the time constant can be

calculated from the cone half-angle by Eq. 54. For the conehalf-angle of 102° determined from the static measurementsand time constant of 350 ps determined from the dynamicalmeasurements, the diffusion constant is Dor=6.6 10−4 ps−1. This diffusion constant for the air-water inter-face should be compared to the one measured in bulk waterof 1.6 10−3 ps−1. This result shows that the reorientation ofthe C314 molecules slows down at the air-water interface bya factor of 2.5.

Now, we examine the surface modified with SDS. SDSforms a monolayer at the air-water interface. The rotationaltime constant was found to increase only by 15% to 383 ps atthe air-water-surfactant interface compared to the time con-stant measured for the pure air-water interface. However, thestatic measurements showed that the equilibrium orienta-tional angle of the C314 molecules with the surfactantchanged substantially to 45° compared the 67° in the absenceof the surfactant.25 As shown in Fig. 3, the cone angle playsan important role in the observed orientational decay time.Using the method outlined above, the orientational diffusionconstant of C314 is found to be 2.6 10−4 ps−1 at the air-water-surfactant interface compared to 6.6 10−4 ps−1 at theair-water interface. This is almost a threefold decrease that isnot reflected in the measured time constants. Here, we fol-lowed the literature and have assumed that the tilt angle doesnot change upon surface modification, that is, the axis of thecone is still normal to the surface. These results are alsophysically intuitive. In the case of the SDS modified surface,the molecular orientational response decays with an experi-mental rate similar to that for the unmodified interface; how-ever, the chromophore traverses a smaller solid angle. This

suggests intuitively that the measured diffusion constantshould be smaller, which is exactly what the formal theoret-ical treatment proves.

The net result of the analysis that yields the orientationaldiffusion constants is to come to conclusions that are verydifferent from those obtained by only comparing the timeconstants. We have determined that the addition of the sur-factant substantially reduces the interfacial diffusion con-stant.

V. CONCLUDING REMARKS

TRSHG and TRSFG experiments have proven to be use-ful tools in investigating the orientational dynamics of mol-ecules at interface. However, little was known about the spe-cific correlation functions measured in the experiments.Some authors interpreted data without accounting for the ef-fects of the restricted rotational diffusion. Others used nu-merical simulation methods that do not permit a generalanalysis of the experiments.

In the work presented here, we calculated the orienta-tional correlation functions for the TRSHG and TRSFG ex-periments. The derivations were performed in general, andthen specific results were obtained for interfacial moleculesthat can be approximated as rigid rods with cylindrical sym-metry. The derivations are valid for resonant experiments,when the probe beam is either in one-photon or two-photonresonance with the electronic or vibrational transition of themolecules under investigation.

The theoretical results show that TRSHG and TRSFGexperiments measure a linear combination of theYl

mlabtY2mlab0 l=1,3 and m=0,1 ,2 orienta-

tional correlation functions. To evaluate the correlation func-tions, we applied the wobbling-in-a-cone model, which de-scribes the reorientational motion of the molecules diffusingin a restricted range of angles, the cone. This model wassuccessfully applied to a number of systems in the bulkphase.31,42,52–55 For the case when the Cv symmetry axis ofthe cone is not the surface normal, the correlation functionwritten in the laboratory coordinates was transformed intothe coordinate system fixed to the conical volume of thediffusion. This transformation permits the general treatmentof any spectroscopic probe at an interface regardless of itsmean orientation angle.

The analytical treatment presented here can be extendedto incorporate any model of diffusion including diffusionmodels with explicit potential functions. As long as aGreen’s function can be solved for analytically or numeri-cally, the correlation functions that have been derived can becalculated and compared to experimental data. Given an ex-perimental configuration, it is now possible to choose theproper correlation functions to calculate and to calculatethem with an appropriate Green’s function.

The general correlation functions derived in this workare also useful for those simulating interfacial molecular sys-tems. Now there exists a clear link between the TRSHG andTRSFG experiments and correlation functions which can bereadily calculated with molecular dynamics simulations.Even when diffusion and other dynamic models fail, molecu-



lar dynamics simulations can be linked directly to experi-mental data through the correlation functions. This linkageenables the validity of simulation models to be tested.

The application of the wobbling-in-a-cone model lead toanalytical expressions of the orientational correlation func-tions, and it was found that they can always be expressed asa single exponential decay to an offset, which is in agree-ment with the experimental findings.25 The analytical expres-sions were used to analyze the results of experiments fromthe literature for a chromophore at the air-water interface andat the air-water-surfactant interface.25 By using both the ex-perimentally determined decay time and cone half-angle, theinterfacial orientational diffusion constants were obtained.The diffusion constants showed that the diffusion constant ofC314 at the air-water is smaller than its bulk value. Addingan interfacial layer of the surfactant SDS reduced the diffu-sion constant by about a factor of 3, which is in contrast tothe conclusions obtained by comparing the experimentallyobserved decay time constants. As illustrated in Figs. 3 and4, the determining the orientational diffusion constant re-quires both the decay time and the cone half-angle in addi-tion to the use of the detailed theory.

It is important to emphasize that a direct comparison ofthe orientational time constants of TRSHG or TRSFG ex-periments to bulk orientational decay constants, or orienta-tional decay constants from interfaces with different meanorientation angles, will not result in a valid comparison oforientational dynamics. We examined uniform angular distri-bution functions in confined solid angles for mathematicalsimplicity. Other models can be used within the same theo-retical framework. The theoretical results also demonstratethe utility of using dynamics as a probe of the angular dis-tribution of chromophores. A significant problem in interfa-cial spectroscopy is the independent determination of bothmean angle and angular distribution. Dynamical measure-ments and their proper interpretation via diffusion modelscan enable information to be gathered not only about themotion of molecules at interfaces but also their averagestructure. Such measurements can provide detailed insightsinto the chemistry and physics of interfacial systems.

Note added in proof. As discussed in the text, for TRSFGthe results are obtained when the transition polarizability ten-sor the derivative of the polarizability tensor taken at theequilibrium geometry is diagonal. The results apply if thevibrational mode is totally symmetric and the transition di-pole moment is along the principle axis of the polarizabilityellipsoid. The results can be readily generalized to vibrationswith symmetries other than totally symmetric by choosing atransition polarizability tensor with the appropriate symme-try. For totally symmetric vibrations, the symmetry of thepolarizability tensor and its derivative are the same.

ACKNOWLEDGMENTS

This research was supported by a grant from the AirForce Office of Scientific Research Grant No. F49620-01-1-0018. D.E.R. thanks the Fannie and John Hertz

Foundation, the Stanford Graduate Fellowship program, andthe National Science Foundation Graduate Research Fellow-ship Program for fellowships.

APPENDIX A: THE CORRELATION FUNCTIONSIN THE WOBBLING-IN-A-CONE MODEL

In this Appendix, we derive the Ylm

tY2m0

Ylm

tY2m0 correlation functions and show that

they are well represented by keeping terms only to the firstdecaying exponential. Using the equations

Ylm

tYlm0

= Yl−m

tYl−m0

=mm

42l + 12l + 1l − m!l − m!

l + m!l + m!

n=1

Cnm exp− n

mnm + 1Dort A1

and

Cnm =

1

Hnm1 − cos o

0

o

d sin Plmcos P

nm

m cos

0

o

d sin Plmcos P

nm

m cos , A2

the Cnm coefficients can be easily calculated. Hn

m denotes theoverlap integrals of the associated Legendre polynomials

Hnm =

0

o

d sin Pn

mm cos P

nm

m cos . A3

We can use that for all o

10 = 0 A4

and

P1

00 cos o = 1. A5

This leads to

H10 = 1 − cos o. A6

At t=0, the correlation function is given as

Ylm

0Ylm0 =

1

21 − cos o0

2

d0

o

d

sin Ylm

0Ylm

0 ,

A7

which is equal to the infinite sum of the exponential coeffi-cients

Ylm

0Ylm0 =

n=1

Cnm. A8



1. The ŠY1m

„Ω„t……Y2m„Ω„0……‹ correlation functions

From Eqs. A1 and A2, the dynamic ensemble aver-ages can be written as

Y10

tY200 =

15

4n=1

An0 exp− n

0n0 + 1Dort

A9

and

Y11

tY210 =

5

8n=1

An1 exp− n

1n1 + 1Dort .

A10

The sum rules can be obtained by setting t=0.

Y10

0Y200 =

15

4n=1

An0

=15

4

5 + 3 cos2osin2 o

161 − cos o

A11

and

Y11

0Y210 =

5

8n=1

An1 =

5

8

3 sin4 o

41 − cos o.

A12

From Eqs. A4–A6, A10 can be calculated analytically

A10 = 1

4cos o1 + cos o2, A13

while the Anmm0 or n2 coefficients can be calculated

only numerically. In Fig. 5, plots of n=1 An

0, A10, A2

0, n=1 An

1,and A1

1 are shown.Figure 5 shows that for o120°,

A20

n=2

An0 =

n=1

An0 − A1

0 =1

4sin2o

2sin2 o A14

and

A11

n=1

An1 =

3 sin4 o

41 − cos o. A15

Because 10=0 and exp−1

010+1Dort=1, only the first two

terms in Eq. A9 contribute meaningfully, and the correla-tion functions can be written as

Y10

tY200 =

15

4A1

0 + A20exp− 2

020 + 1Dort

A16

and

Y11

tY210 =

5

8A1

1 exp− 111

1 + 1Dort .

A17

2. The ŠY3m

„Ω„t……Y2m„Ω„0……‹ correlation

functions

Let us write the dynamic ensemble averages as

Y30

tY200 =

35

4n=1

Bn0 exp− n

0n0 + 1Dort ,

A18

Y31

tY210 =

35

242n=1

Bn1 exp− n

1n1 + 1Dort ,

A19

and

Y32

tY220 =

7

96n=1

Bn2 exp− n

2n2 + 1Dort .

A20

Following the above applied procedure, first we find the sumrules are

Y30

0Y200

=35

4n=1

Bn0 =

1

32cos2o

212 cos 2o + 53 + cos 4o ,

A21

Y31

0Y210 =

35

242n=1

Bn1

=3

8

7 + 5 cos 2osin4 o

1 − cos o, A22

and

0 20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0

( )0 0 01 2

1

& nn

A A A∞

=

+ ∑

1 11

1

& nn

A A∞

=∑

θo (degrees)

A nm

FIG. 5. The Anm parameters of the Y1

mtY2

m0 correlation func-tions. The solid lines are calculated from Eqs. A11 and A12 for the sumrules. The dashed lines are calculated for A1

0+A20 and A1

1 numerically. Seedetails in text.



Y32

0Y220 =

7

96n=1

Bn2 =

15

2

sin6 o

1 − cos o.

A23

Again, from Eqs. A4–A6 B10 can be calculated analyti-

cally

B10 =

1

16cos o1 + cos o25 cos2 o − 1 , A24

while the Bnmm0 or n2 coefficients can be calculated

only numerically. In Fig. 6, plots of n=1 Bn

0, B10, B2

0; n=1 Bn

1,B1

1, B21; and n=1

Bn2, B1

2 are shown. Figure 6 shows that foro110°,

B20

n=2

Bn0 =

n=1

Bn0 − B1

0

=1

4cos2o

29 + 10 cos o + 5 cos 2osin4o

2

A25

and

B12

n=1

Bn1 =

15

2

sin6 o

1 − cos o. A26

Using that 10=0 and exp−1

010+1Dort=1 and approxima-

tions above, the correlation functions can be written as

Y30

tY200 =

35

4B1

0 + B20exp− 2

020 + 1Dort ,

A27

Y31

tY210 =

35

242B1

1 exp− 111

1 + 1Dort,

for o 70 ° only , A28

and

Y32

tY220 =

7

96B1

2 exp− 121

2 + 1Dort .

for o 110 ° only . A29

APPENDIX B: ALGEBRAIC TRANSFORMATIONSOF THE SPHERICAL HARMONIC FUNCTIONS

The evaluation of the orientational correlation functionsrequires algebraic transformation of the spherical harmonicrepresentations of the field-matter interactions. Using simplerelationships between spherical harmonics of different order,those representations can always be written as the sum of aconstant and spherical harmonics of the order of one to three.The pump field can be polarized along any of the Cartesiancoordinate axis, and we also consider the case when it propa-gates along the Z axis and is circularly polarized in the XYplane. Because the pump interaction is approximated as in-stantaneous, the pump field-matter interaction can be writtenfor different polarizations as

iZ2 =1

3+

4

3

5Y2

0ti , B1

iY2 =1

3−

2

3

5Y2

0ti −2

15Y−2

2 ti

+ Y22ti , B2

iX2 =1

3−

2

3

5Y2

0ti +2

15Y−2

2 ti

+ Y22ti , B3

and

iCIRC2 =2

3−

4

3

5Y2

0ti . B4

In the probe interactions, only the polarization combinationsthat are allowed by the symmetry of the surface are consid-ered.

iZZiZ = ZiZiZ

=2

15

315 + 4Y1

0ti

+4

5

7Y1

0ti , B5

0 20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0

0 0 01 2

1

( ) & nn

B B B∞

=

+ ∑

θo (degrees)

B nm

0 20 40 60 80 100 1200

2

4

6

8

10

1 11

1

& nn

B B∞

=∑

2 21

1

& nn

B B∞

=∑

θo (degrees)

B nm

(b)

(a)

FIG. 6. a and b The Bnm parameters of the Y3

mtY2

m0 corre-lation functions. The solid lines are calculated from Eqs. A21 and A22for the sum rules. The dashed lines are calculated for B1

0+B20, B1

1, and B12

numerically. See details in text.



iZYiY = YiYiZ

=2

15

315 − 2Y1

0ti

− 2

105Y3

−2ti + Y32ti

−2

5

7Y3

0ti , B6

iZXiX = XiXiZ

=2

15

315 − 2Y1

0ti

+ 2

105Y3

−2ti + Y32ti

−2

5

7Y3

0ti B7

iYYiZ = iYZiY

= YiZiY

= ZiYiY

=2

5

3Y1

0ti − 2

105Y3

−2ti

+ Y32ti −

2

5

7Y3

0ti B8

iXXiZ = iXZiX

= XiZiX

= ZiXiX

=2

5

3Y1

0ti + 2

105Y3

−2ti

+ Y32ti −

2

5

7Y3

0ti . B9

1 G. A. Somorjai, Introduction to Surface Chemistry and Catalysis Wiley,New York, 1994.

2 G. A. Somorjai, Annu. Rev. Phys. Chem. 45, 721 1994.3 T. Kakuchi, in Interfacial Catalysis, edited by A. G. Volkov Dekker,New York, 2003.

4 P. Moriarty, Rep. Prog. Phys. 64, 297 2001.5 R. G. Quiller, L. Benz, J. Haubrich, M. E. Colling, and C. M. Friend, J.Phys. Chem. C 113, 2063 2009.

6 C. R. Usher, A. E. Michel, and V. H. Grassian, Chem. Rev. Washington,D.C. 103, 4883 2003.

7 S. McLaughlin, Annu. Rev. Biophys. Biophys. Chem. 18, 113 1989.8 A. G. Volkov, D. W. Deamer, D. L. Tanelian, and V. S. Markin, LiquidInterfaces in Chemistry and Biology Wiley, New York, 1998.

9 Y. R. Shen, The Principles of Nonlinear Optics Wiley, New York, 1984.10 R. W. Boyd, Nonlinear Optics, 3rd ed. Elsevier, New York, 2008.11 H.-F. Wang, W. Gan, R. Lu, Y. Rao, and B.-H. Wu, Int. Rev. Phys. Chem.

24, 191 2005.12 X. Zhuang, P. B. Miranda, D. Kim, and Y. R. Shen, Phys. Rev. B 59,

12632 1999.13 Y. Rao, M. Comstock, and K. B. Eisenthal, J. Phys. Chem. B 110, 1727

2006.14 G. A. Somorjai and J. Y. Park, Surf. Sci. 603, 1293 2009.15 Y. R. Shen and V. Ostroverkhov, Chem. Rev. Washington, D.C. 106,

1140 2006.16 G. A. Somorjai and J. Y. Park, Chem. Soc. Rev. 37, 2155 2008.17 K. M. Bratlie, K. Komvopoulos, and G. A. Somorjai, J. Phys. Chem. C

112, 11865 2008.18 X. Wei and Y. R. Shen, Phys. Rev. Lett. 86, 4799 2001.19 A. Castro, E. V. Sitzmann, D. Zhang, and K. B. Eisenthal, J. Phys. Chem.

95, 6752 1991.20 R. Antoine, A. A. Tamburello-Luca, P. Hébert, P. F. Brevet, and H. H.

Girault, Chem. Phys. Lett. 288, 138 1998.21 K. Sekiguchi, S. Yamaguchi, and T. Tahara, J. Chem. Phys. 128, 114715

2008.22 Y. Rao, D. Song, N. J. Turro, and K. B. Eisenthal, J. Phys. Chem. B 112,

13572 2008.23 J. A. McGuire and Y. R. Shen, Science 313, 1945 2006.24 D. Zimdars, J. I. Dadap, K. B. Eisenthal, and T. F. Heinz, J. Phys. Chem.

B 103, 3425 1999.25 K. T. Nguyen, X. Shang, and K. B. Eisenthal, J. Phys. Chem. B 110,

19788 2006.26 M. J. Wirth and J. D. Burbage, J. Phys. Chem. 96, 9022 1992.27 X. Shang, K. Nguyen, Y. Rao, and K. B. Eisenthal, J. Phys. Chem. C

112, 20375 2008.28 H.-K. Nienhuys and M. Bonn, J. Phys. Chem. B 113, 7564 2009.29 M. P. Warchol and W. E. Vaughan, Adv. Mol. Relax. Interact. Processes

13, 317 1978.30 C. C. Wang and R. Pecora, J. Chem. Phys. 72, 5333 1980.31 A. Tokmakoff and M. D. Fayer, J. Chem. Phys. 103, 2810 1995.32 B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications

to Chemistry, Biology, and Physics Dover, Mineola, 2000.33 A. Szabo, J. Chem. Phys. 81, 150 1984.34 A. Tokmakoff, J. Chem. Phys. 105, 1 1996.35 T. Tao, Biopolymers 8, 609 1969.36 P. J. W. Debye, Polar Molecules Dover, New York, 1945.37 A. Carrington and A. D. McLachlan, Introduction to Magnetic Resonance

with Applications to Chemistry and Chemical Physics Harper & Row,New York, 1967.

38 A. Tokmakoff, R. S. Urdahl, D. Zimdars, R. S. Francis, A. S. Kwok, andM. D. Fayer, J. Chem. Phys. 102, 3919 1995.

39 J. M. Beechem and L. Brand, Annu. Rev. Biochem. 54, 43 1985.40 R. E. Dale, S. C. Hopkins, U. A. an der Heide, T. Marszalek, M. Irving,

and Y. E. Goldman, Biophys. J. 76, 1606 1999.41 H. Vogel and F. Jahnig, Proc. Natl. Acad. Sci. U.S.A. 82, 2029 1985.42 I. R. Piletic, D. E. Moilanen, D. B. Spry, N. E. Levinger, and M. D.

Fayer, J. Phys. Chem. A 110, 4985 2006.43 A. Szabo, J. Chem. Phys. 72, 4620 1980.44 D. E. Moilanen, E. E. Fenn, Y. S. Lin, J. L. Skinner, B. Bagchi, and M.

D. Fayer, Proc. Natl. Acad. Sci. U.S.A. 105, 5295 2008.45 S. Mukamel, Principles of Nonlinear Optical Spectroscopy Oxford Uni-

versity Press, New York, 1995.46 S. Mukamel and R. F. Loring, J. Opt. Soc. Am. B 3, 595 1986.47 A. Tokmakoff, J. Chem. Phys. 105, 13 1996.48 S. Park, K. Kwak, and M. D. Fayer, Laser Phys. Lett. 4, 704 2007.49 M. Cho, G. R. Fleming, and S. Mukamel, J. Chem. Phys. 98, 5314

1993.50 G. R. Fleming, Chemical Applications of Ultrafast Spectroscopy Oxford

University Press, New York, 1986.51 X. Wei, P. B. Miranda, C. Zhang, and Y. R. Shen, Phys. Rev. B 66,

085401 2002.52 A. Tokmakoff, A. S. Kwok, R. S. Urdahl, R. S. Francis, and M. D. Fayer,

Chem. Phys. Lett. 234, 289 1995.53 D. E. Moilanen, E. E. Fenn, D. Wong, and M. D. Fayer, J. Am. Chem.

Soc. 131, 8318 2009.54 D. E. Moilanen, D. Wong, D. E. Rosenfeld, E. E. Fenn, and M. D. Fayer,

Proc. Natl. Acad. Sci. U.S.A. 106, 375 2009.55 E. E. Fenn, D. E. Moilanen, N. E. Levinger, and M. D. Fayer, J. Am.

Chem. Soc. 131, 5530 2009.56 B. Pal, Bull. Calcutta Math. Soc. 9, 85 1919.57 B. Pal, Bull. Calcutta Math. Soc. 10, 187 1919.58 R. N. Zare, Angular Momentum: Understanding Spatial Aspects in

Chemistry and Physics Wiley, New York, 1988.59 M.-L. Horng, J. A. Gardecki, and M. Maroncelli, J. Phys. Chem. A 101,

1030 1997.



http://dx.doi.org/10.1146/annurev.pc.45.100194.003445

http://dx.doi.org/10.1088/0034-4885/64/3/201

http://dx.doi.org/10.1021/jp806900j


http://dx.doi.org/10.1021/cr020657y

http://dx.doi.org/10.1021/cr020657y

http://dx.doi.org/10.1146/annurev.bb.18.060189.000553

http://dx.doi.org/10.1080/01442350500225894

http://dx.doi.org/10.1103/PhysRevB.59.12632

http://dx.doi.org/10.1021/jp055340r

http://dx.doi.org/10.1016/j.susc.2008.08.030

http://dx.doi.org/10.1021/cr040377d

http://dx.doi.org/10.1039/b719148k

http://dx.doi.org/10.1021/jp801583q

http://dx.doi.org/10.1103/PhysRevLett.86.4799

http://dx.doi.org/10.1021/j100171a003

http://dx.doi.org/10.1016/S0009-2614(98)00239-5

http://dx.doi.org/10.1063/1.2841023

http://dx.doi.org/10.1021/jp802499w

http://dx.doi.org/10.1126/science.1131536

http://dx.doi.org/10.1021/jp984468o

http://dx.doi.org/10.1021/jp984468o


http://dx.doi.org/10.1021/j100201a060

http://dx.doi.org/10.1021/jp807273v

http://dx.doi.org/10.1021/jp810927s

http://dx.doi.org/10.1016/0378-4487(78)80053-3

http://dx.doi.org/10.1063/1.439024

http://dx.doi.org/10.1063/1.470517

http://dx.doi.org/10.1063/1.447378

http://dx.doi.org/10.1063/1.471856

http://dx.doi.org/10.1002/bip.1969.360080505

http://dx.doi.org/10.1063/1.468568

http://dx.doi.org/10.1146/annurev.bi.54.070185.000355

http://dx.doi.org/10.1016/S0006-3495(99)77320-0

http://dx.doi.org/10.1073/pnas.82.7.2029

http://dx.doi.org/10.1021/jp061065c

http://dx.doi.org/10.1063/1.439704

http://dx.doi.org/10.1073/pnas.0801554105

http://dx.doi.org/10.1364/JOSAB.3.000595

http://dx.doi.org/10.1063/1.471859

http://dx.doi.org/10.1002/lapl.200710046

http://dx.doi.org/10.1063/1.464931

http://dx.doi.org/10.1103/PhysRevB.66.085401

http://dx.doi.org/10.1016/0009-2614(95)00068-F

http://dx.doi.org/10.1021/ja901950b

http://dx.doi.org/10.1021/ja901950b

http://dx.doi.org/10.1073/pnas.0811489106

http://dx.doi.org/10.1021/ja809261d

http://dx.doi.org/10.1021/ja809261d

http://dx.doi.org/10.1021/jp962921v

60 The orientational relaxation time constant of 106 ps is a personal com-munication from Professor K. B. Eisenthal, Department of Chemistry,Columbia University 7 Feb. 2010 and has been verified by independentmeasurement in our laboratory. A different value was published by K. B.

Eisenthal and co-workers in Ref. 27.61 D. A. Pantano and D. Laria, J. Phys. Chem. B 107, 2971 2003.62 D. A. Pantano, M. T. Sonoda, M. S. Skaf, and D. Laria, J. Phys. Chem. B

109, 7365 2005.



http://dx.doi.org/10.1021/jp021306q

http://dx.doi.org/10.1021/jp045687e

Theory of interfacial orientational relaxation ...

Documents

Transcript of Theory of interfacial orientational relaxation ...