空气/水界面中和频产生（SFG）振动光谱,偏振特性,实验装置分析检测方案(其它光谱仪)

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2 3 Polarization and Experimental Configuration Analysis of Sum Frequency GenerationVibrational Spectra of Air/Water Interface Wei Gant， Dan Wut， Zhen Zhangt， Ran-ran Fengt， and Hong-fei Wang* State Key Laboratory of Molecular Reaction Dynamics， Institute of Chemistry， the Chinese Academy of Sciences， Beijing， China， 100080 (Dated： February 2， 2008) Here we report a detailed study on spectroscopy， structure and dynamics of water molecules atair/water interface， investigated with Sum Frequency Generation Vibrational Spectroscopy (SFG-VS). Quantitative polarization and experimental configuration analysis of the SFG data in differ-ent polarizations with four sets of experimental configurations can shed new lights on our presentunderstanding of the air/water interface. Firstly， we concluded that the motion of the interfacialwater molecules can only be in a limited angular range， instead rapidly varying over a broad angularrange in the vibrational relaxation time suggested previously. Secondly， because different vibrationalmodes of different molecular species at the interface has different symmetry properties， polarizationand symmetry analysis of the SFG-VS spectral features can help assignment of the SFG-VS spectrapeaks to different interfacial species. These analysis concluded that the narrow 3693cm-and broad3550cmpeaks belong to Coou symmetry， while the broad 3250cmand 3450cmpeaks belongto the symmetric stretching modes with C2v symmetry. Thus， the 3693cm peak is assigned tothe free OH， the 3550cm peak is assigned to the single hydrogen bonded OH stretching mode，and the 3250cm-and 3450cm-peaks are assigned to interfacial water molecules as two hydrogendonors for hydrogen bonding (with C2u symmetry)， respectively. Thirdly， analysis of the SFG-VSspectra concluded that the singly hydrogen bonded water molecules at the air/water interface havetheir dipole vector direct almost parallel to the interface， and is with a very narrow orientationaldistribution. The doubly hydrogen bond donor water molecules have their dipole vector point awayfrom the liquid phase. oou寸oooInterfaces of water are the most important subjects notonly because water is widely involved in physical， chem-ical， environmental as well as biological processes， butalso because water is so far the most mysteries moleculein the universe.1.2.3.4 Among them， air/water interfacehas been intensively investigated theoretically or experi-mentally over the last decades. Spectroscopy， molecularstructure and dynamics at air/water interface is stud-ied with theoretical analysis such as ab initio calculationormmolecular dynamics simulation，5.6.2.8.9.10，11，12.13 or ex-perimental techniques such as X-ray reflection，，15 Stim-ulated Raman Scattering (SRS)，6 Near-edge X-Ray Ad-asorption Fine Structure (NEXAFS)，- Second HarmonicoGeneration (SHG)18.19 as well as Sum Frequency Gener-ation，etc.20.2122.23.24.25.26.27.28 Among these expemen-tal techniques， Second Harmonic Generation and SumFrequency Generation are the most important methodsfor molecular interface studies because of their surfacesensitiitviivity and specificity.29，30，31，32，33，34，35，36 With theseinvestigations， the properties of the water molecules atthe interface， such as the surface density， surface struc-ture， surface potential as well as surface dynamics， havebeen intensively discussed. TAlso Graduate School of the Chinese Academy of Sciences*Author to whom correspondence should be addressed. E-mail：hongfei@mrdlab.icas.ac.cn. Tel. 86-10-62555347，Fax 86-10-62563167. However， conclusions on the surface molecule speciesat air/water interface are still under discussion.17.20.23.25With SFG-VS experimental studies， the following inter-facial water species have been reported in literatures，namely， water molecules straddle at the interface withone OH bond hydrogen bonded to neighboring moleculesin liquid phase (singly bonded OH) and another OHbond free from hydrogen bonding (free OH) in gasphase：2223，24，37. wwater molecules with both OH bondssymmetrically hydrogen bonded in a tetrahedral network(ice-like and liquid-like structures)；22.23.24.37 and watermolecules in gas phase with both OH bonds not hy-drogen bonded pointing into the liquid phase.23.26WithNEXAFS measurement and ab initio molecular dynam-ics simulation， water molecules with both OH bondsnot hydrogen bonded pointing out of the interface wasalso proposed.13.17 The latter case is particularly contro-versial because NEXAFS is not strictly a surface spe-cific technique. With polarization SFG-VS measure-ment， Wei et al. discussed the absence of SFG spectrain some polarization combinations and proposed an ex-planation through fast orientational motion in a broadrange of about 102° in a time scale comparable or lessthan 0.5 ps.25 However， puzzle still remains becausesome of the experimental studies suggests ordered andslow dynamics for interfaces of hydrogen bonding liq-uids， while some experimental investigations suggesteda more dynamic and less ordered picture for the liq-uid interfaces， air/water interface included.3 In addi-tion， whether the surface orientation relaxation is fastor slow than the bulk water molecules is also an issueunder discussion in the recent literatures.12.38，39 Besides SHG and SFG-VS experimental studies，33.40 Structureand dynamics of water molecules at the air/water inter-face have also been intensively discussed with theoreticalsimulations.2.6..8.9，10，1，12，13 Even though with so mefforts and progresses both by experimentalists and the-oreticians， our detailed understanding of air/water inter-face is still limited. Just as indicated by B. C. Garrettrecently，-‘...(direct) experiments are difficult to performbecause the liquid interface is disordered， dynamic， andsmall (typically only a few molecules wide) relative to thebulk’. Actually， direct measurement of the liquid interfaceis not as difficult as suggested as above. IIt has beenknown that along with SHG， SFG-VS can provide directmeasurement on liquid interface no other technique canmatch.33.40 As pointed out by Miranda and Shen， ‘SFG iscurrently the only technique that can yield a vibrationalspectrum for a neat liquid interface’33 In fact， with theadvances of ultrafast laser and detection technology inth past decade and especially recent few years，42.43.44.45particularly with commercial systems designed for SFG-VS measurement，46 SFG-VS， as well as SHG， experi-ments have come from easier to routine.47 The real dif-ficulty lies on the fact that quantitative analysis andinterpretation of the SFG-VS， as well as SHG， datahad been not as well developed and widely performedlintil recently25.35.48，49，50.51，52，53.54，55，56，57.58Therefore.conclusions in many previous reports on the investiga-tions of air/water interface， as well as other liquid inter-faces， with SFG-VS are subjected to different interpreta-tions. As We haveedemonstratedma seriesoffre-cent publications.systematicallyyquantitative treat-ment toSFG-VSdataisnot only possible，，butalso very effective for obtaining detailed spectroscopic，structuraland thermodynamicproperties of liquidinterfaces.49.50.51，52，53，54，55，56.57.58 In these works， we notonly developed methodology for quantitative polarizationand experimental configuration analysis in SFG-VS andSHG， we also tested accuracy and sensitivity of some ofthe methodology. We have applied them to elucidatedthe anti-parallel double layered structure and thermo-dynamics of some organic liquid aqueous solution in-terfaces. In addition， we also demonstrated that a setof polarization selection rules (or guidelines) in SFG-VS can be developed for vibrational spectrum assign-ment through symmetry analysis of the SFG-VS spec-tral features.. This latter approach is extremely usefulfor discerning complex SFG-VS spectrum with unidenti-fied or controversial assignments. Recently， based on po-larization analysis， Ostroverkhov et al. demonstrated aphase-sensitive interference analysis of SFG polarizationspectra of 59 Wwater/quartz interface. With these develop-ment， in this report we intend to apply these analysismethodologies to the study of air/water interface. Inthiswork.we examined SFG-VSsspectraatair/water interface measured in different polarizationsunder four experimental configurations with polarization analysis method and experimental configuration analy-sis.3.With these analysis， detailed new information areobtained for understanding of the spectroscopy， struc-ture and dynamics of the air/water interface. In thefollowing sections， after a brief introduction of the theo-retical background and experimental conditions， we firstdiscuss the motion of the interfacial water molecules atthe air/water interface， which was previously suggestedexperiencing rapidly motion over a broad angular rangein the vibrational relaxation time； then we use polar-ization and symmetry analysis of the SFG-VS spectralfeatures for assignment of the SFG-VS spectra peaks； inthe end， we shall discuss the structure and orientation ofthe water molecules at the air/water interface. II..POLARIZATION AND EXPERIMENTALCONFIGURATION ANALYSIS IN SFG-VS Quantitative polarization analysis and experimentalconfiguration analysis can provide rich and detailed infor-mation of spectroscopy， structure and dynamics of molec-ylar interfaces25.35.49，50.1.551.15.35 3G eGneenrearlallyly， the SFG inten-sity in the reflective direction is，35.50 in which w， wi and w2 are the frequencies of the SFGsignal， visible and IR laser beam， respectively. ni(wi) isthe refractive index of bulk medium i at frequency wi，and n'(wi) is the effective refractive index of the inter-face layer at wi.. is the incident or reflection anglefrom interface normal of the ith light beams； I(wi) is theintensity of t(2)he SFG signal or the input laser beam. Xeffisi1sCs the effective second order susceptibility for an inter-face. The notations and the experiment geometry havebeen described in detail previously.35.50 (2)xXeff， for the four generally used independent polar-ization combinations can be deduced from the 7 nonzeromacroscopic susceptibility tensors for an achiral rotation-ally isotropic interface (Coou).35.50 It is so defined that the ry plane in the laboratory co-ordinates system A(c，y，z) is the plane of interface； all the light beams propagate in the xz plane； p denotes thepolarization of the optical field in the az plane， with zas the surface normal， while s the polarization perpen-dicular to the az plane. The consecutive superscript，such as ssp， represents the following polarization combi-nations： SFG signal s polarized， visible beam s polarized，IR beam p polarized， and so forth. Lii (i=a，y，z) is theFresnel coefficient determined by the refractive indexes ofthe two bulk phase and the interface layer， and the inci-dent and reflected angles.35.50Xijktensors are related tothe microscopic hyperpolarizability tensor Bii'k of themolecules in the molecular coordinates system X'(a，b， c)through the ensemble average over all possible molecularorientations.35.50 where Rxx(0，，v) is the matrix element of the Eu-ler rotational transformation matrix from the molecu-lar coordination X’(a，b，c) to the laboratory coordination(z，y，z)； Firiw is the microscopic (molecular) hyperpo-larizability tensor.52.55.60 Here N is the molecular num-ber density at the interface. (A) represents orientationalaverage of property A(0，d，) over the orientational dis-tribution function f(0，o，p). For SFG-VS， 3(2) is IR frequency (w2) dependent， Thus， Xink can be expressed into， Therefore， SFG-VS measures the vibrational spec-troscopy of molecular interfaces. For dielectric interfaces，such as liquid interfaces， the non-resonant term BNR，ii'kor xyRiik is generally negligible compare with the reso-nant terms. Recently， we have found that the following formulationis very effective in quantitative polarization and orienta-tion analysis of SFG and SHG data. It can be generallyshown that in surface SFG and SHG for an interface withorientational order， the effective second order susceptibil-ity xerf can be simplified into the following form.4 r(0) is called the orientational field functional， which con-tains all molecular orientational information at a givenSFG experimental configuration； while the dimensionlessparameter c is called the general orientational parameter，which determines the orientational response r(0) to themolecular orientation angle 0； and d is the susceptibilitystrength factor， which is a constant in a certain experi-mental polarization configuration with a given molecularsystem. The d and c values are both functions of the re-lated Fresnel coefficients including the refractive index ofthe interface and the bulk phases， and the experimentalgeometry. The key for quantitative analysis is that both d andc can be explicitly derived from the expressions of theXeff in relationship to the macroscopic susceptibility andmicroscopic (molecular) hyperpolarizability tensors fora particular molecular vibrational modes，50.52 as shownfor the water molecules with C2v symmetry in the ap-pendix. With the parameters c and d， the polariza-tion dependence and the orientation dependence of theSFG/SHG signal for a certain interface at certain ex-perimental configuration can be analyzed and calculatedwith clear physical picture on molecular orientation andorientational distribution. Reciprocally， information onthe molecular symmetry， molecular orientation and dy-namics can be obtained from the analysis on the SFGintensity relationships measured in different polarizationcombinations and experimental configurations.50.51.52.53 The orientational average in Eq3 is only the staticaverage on molecular orientations， without consideringfast molecular motion effects. Recently Wei et al. dis-cussed the fast and slow limit of the time average overorientational motion for Xeff， and they also applied thistreatment to analysis the polarization dependence of SFGmeasurement of the OH stretching vibrational spectra forthe air/water interface.25 In the fast motion limit， theorientational motion is faster than the vibrational relax-ation time scale 1/Iq of the qth vibrational mode； whilein the slow motion limit， the orientational motion is muchslower than 1/Tq According to Wei et al.，25 the slow motion limit gives， while the fast motiongives， in which Rxx(t) =A(t) is the time-dependent di-rection Euler transformation matrix from X(a，b，c) toA(z，y，z) coordinates system. Because of the molecularorientational motion， the molecular coordinates X'(a，b，c)is time-dependent. Eq.B is equivalent to Eq d， which isobtained by insertion of Eq.linto Eq.B. The details of the laser system has been described inour previous reports.50.56.57 Briefly， the 10Hz and 23 pi-cosecond SFG spectrometer laser system (EKSPLA) isin a co-propagating configuration. The efficiency of thedetection system has been improved for the weak SFGsignal of air/water interface. A high-gain low-noise pho-tomultiplier (Hamamatsu， PMT-R585) and a two chan-nel Boxcar average system (Stanford Research Systems)are integrated into the EKSPLA system. The voltageof R585 was 1300V in the measurement for air/waterinterface， and 900V for the Z-cut quartz surface. Thewavelength of the visible is fixed at 532nm and the fullrange of the IR tunability is 1000cm-1 to 4300cm-1.The specified spectral resolution of this SFG spectrome-ter is <6cm-in the whole IR range， and about 2cm-around 3000cm-1. Each scan was with a 5cm-1 incre-ment and was averaged over 300 laser pulses per point.Each spectrum has been repeated for at least severaltimes. Moreover， for sps polarization， each spectrum hasbeen repeated for more than a dozen times and averaged.The energy of visible beam is typically less than 300pJand that of IR beam less than 150pJ around 3000cm-and 3700cm-1， and less than 100uJ in the region in be-tween. These are comparable to literature reported val-ues for measurement of air/water interface.20 All mea-surements were carried out at controlled room temper-ature (22.0±0.5°C) and humidity (40%). The sampleused was ultrapure water from standard Millipore treat-ment (18.2 MQ·cm). The whole experimental setup onthe optical table was covered in a plastic housing to re-duce the air flow. No detectable evaporation effect wasobserved for SFG spectrum during each scan. The normalization procedure of the SFG signal in dif-ferent experimental configurations need to be specificallydiscussed. The detail of the normalization procedure fora single experimental configuration was presented in XingWei’s Ph.D. dissertation.2 However， the difference of co-herent length and Fresnel factors with different incidentangles in the quartz SFG signal measurements has to becorrected when comparing SFG signal in different exper-imental configurations..Therefore， the measured spec-trum is firstly normalized with the energy of the incidentlaser beams， and then normalized to the SFG signal ofZ-cut quartz (also normalized by the energy of the in-cident lasers). Then it times with a converting factorbetween different experimental configurations. This fac-tor contains the influence of the coherent length of Z-cutquartz，2 the Fresnel coefficients，2 the Xijk value for Z-cut quartz， and the factor secp for each experimentalconfiguration. Therefore， the end result is directly pro-portional to the SFG intensity in Eq.. If the spectrum inFig.is divided by the factor sec p and the factor of thePMT efficiency between 1300V and 900V， which is de-termined as 24.1 in our detection system， and then timesthe unit factor 1x 10-40v4m-2 which we left out forsimplicity of graph presentation， it will give the value for |x 2. For example， the peak at about 3700cm-1 in thessp spectra of Config.2 in Fig.M is about 0.23 unit. Afterabove conversion it gives lxerfl?=4.7×10-40v4m-2，matching satisfactorily with the reported value for lessthan 10% difference.25 Even though the normalized intensities are generallyconsistent with each other， there can be possibly othersources of errors when intensities in different experimen-tal configurations need to be compared. Because the vis-ible and IR beams have different coherent lengthes in theZ-cut crystal， and because these coherent lengthes varywith different experimental incident angles， one of themost likely error might come from the different focus-ing parameters with different beam overlapping qualityof the visible and IR beams in the Z-cut quartz crys-tal with different experimental configurations.There-fore， quantitative comparison of the SFG spectral inten-sities in different polarizations with the same experimen-tal configuration can be more accurate than comparisonintensities between different experimental configurations.Even though the latter is a good solution to reduce suchrelative error associated with different experimental con-figurations need to be developed. IV. RESULTS AND DISCUSSION A... IPolarization SFG Spectra of the air/waterinterface Firstly we would like to present the polarization SFGspectra of the air/water interface measured in four dif-ferent experimental configurations. We have demonstrated recently that the change of theSFG spectra in different polarizations by varying the ex-perimental configurations can be used for quantitativepo zation analysis and orientational analysis.2.53 Herewe present in Fig.the SFG spectra in the ssp， ppp andsps polarizations on the air/water interface at four ex-perimental configurations with different incident anglesfor the visible and IR laser beams. They are， Config.1：Visible=39°， IR=55°； Config.2：：Visible=45°， IR=55°；Config.3：：Visible=48°， IR=57°； Config.4：：Visible=63°，IR=55°. There are four apparent peaks can be identified inthe SFG spectra in Fig..They are around 3700cm-，3550cm-1， 3450cm-1 and 3250cm-1， respectively. The3700cm-1，3450cm-and 3250cm--1peaks has beenextensively discussed in the SFG literature.21.22.23.24.37However， the 3550cm-peak has been observed， but notyet clearly identified or assigned.25 The results of globalfit of these spectra with four Lorentzian peaks in Eq.dare listed in Table From the fitting results we can seethat the peak bandwidths of the 3550cm-1， 3450cm-1and 3250cmpeaks are 77±11cm-1，103±7cm-and89 ±9cm-， respectively. Such broad bandwidths indi-cate that they all belong to different hydrogen bonded O-H stretching vibrational modes. However， the bandwidth Wavenumber (cm FIG. 1： SFG spectra of air/water interface in different polarization combination and experimental configurations. All spectraare normalized to the same scale. The solid lines are globally fitted curves with Lorentzian line shape function in Eq. d. Notethe different error bars for graphs in different scales. of the 3693cm-peak width is only 17cm-1， consistentwith the symmetric stretching (ss) vibrational mode ofthe free O-H bond.20 The signs in Table contain theinformation of the relative phase and interference effectsof the different vibrational modes. Here the phase of the3693cm-1 peak is held positive in each fit. Altering therelative phases of the peaks on the same spectrum cannot give a reasonable fit. Because we used global fittingwith all the spectra， these relative phases can be deter-mined accurately. They can be used to determine thesymmetry properties of each vibrational mode in SectionIV.C. According to Eq the ssp spectra in different experi-mental configurations should have the same features from the Xyyz term. As shown in Figall ssp curves over-lap quit well when normalized to the 3693cm-peak.Calculation of the Fresnel factors with different incidentangles can quantitatively explain the relative intensitiesin all four configurations.3 Because the SFG spectral in-tensity from the air water interface in the OH region isusually several times smaller than that of the C-H regionfrom other air/liquid interfaces， the air/water interfaceSFG spectra are usually very hard to measure experimen-tally. Therefore， the well overlapping of the ssp spectrain different experimental configurations is a proof for thequality of our SFG-VS data. Furthermore， the spectra weobtained agree very well with these in the literatures.25.64 In principle， the sps spectra in different experimental TABLE I： The fitting results of the SFG spectra at air/watrerinterface. The spectra are fitted with Lorentzian line shapefunction as Eq.6'The peak position of the vibrational modeswq， the peak width Tq and the oscillator strength factorXeffoijk of the vibrational modes are listed. The first col-umn is the fitted value for xNR，effijk. The relative error infitting of sps is larger because of the small signal strength forsps spectra. wq(cm-) 3281±5 3446±3 3536±6 3693±1 Tg(cm- ) 89±9 103±7 77±11 17±1 sSp0.17 -6.7±0.6 -20.1±1.3-5.2±1.26.8±0.2 Config.1 ppp -0.04 3.2±0.5 2.6±0.7 5.0±0.4 1.1±0.1 sps -0.01 -0.1±0.1 -0.2十0.1 -3.5±0.50.9±0.2 ssp0.19 -8.3±0.6-24.0±3.5-5.0±3.58.5±0.1 Config.2 ppp -0.02 1.1±0.5 0.0±0.8 6.9±0.32.4±0.6 sps0.02 -0.1土0.1 -0.3±0.1 -4.5±0.51.6±0.1 ssp 0.22 -10.1±0.7-22.8±1.5-7.0±2.0 8.8±0.2 Config.3 ppp -0.01 2.4±0.6 0.9±0.7 6.6±0.32.8±0.1 sps0.01 -0.2±0.1 -0.3±0.1 -3.4±0.71.4±0.2 ssp0.21 -8.8±0.7 -23.3±1.4-5.0±1.59.2±0.2 Config.4 ppp 0.15 -1.0±0.8 -3.0±1.3 -9.0±0.79.3±0.2 sps 0.01 -0.2±0.1 -0.4±0.1-6.6±0.83.1±0.2 configurations should also overlap with each other whennormalized. However， consistent with the calculationsof the corresponding Fresnel factors， the sps signal levelfor Config. 1， 2 and 3 are very close to the noise level，and features in the sps spectra can not be clearly iden-tified except for the spectra of Config.4. Therefore， suchnormalization and comparison for sps spectra is not asmeaningful as the ssp spectra. Different from the ssp and sps spectra， the features inthe ppp spectra in Fig.l changed drastically with differ-ent experimental configurations. This is because that theppp spectra is determined by combination of four differ-ent Xijk tensors. Detailed polarization analysis and ex-perimental configuration analysis of these changes in theppp spectra can provide symmetry properties for eachspectral features， as well as orientation and structure in-formation of the interfacial molecular groups， as shall beshown later.50.51，52 We shall show that analysis of theppp spectra in different experimental configurations isvery informative. However， this advantage of ppp spec-tra analysis has not been well utilized in the previousliteratures. B. Orientation and Motion of the Free OH Bond Now with the knowledge of the SFG vibrational spec-tra of the air/water interface， we can discuss the orien-tation and motion of the free O-H bond at the air/waterinterface.l1 FIG. 2： Overlap of the normalized ssp spectra of the air/waterinterface in different experimental configurations. The sharp peak around 3700cm-1 was generally ac-cepted as the free OH bond protruding out of the liquidwater，20.23.25.66 and it has been treated with Coou sym-metry in polarization analysis.20.25 Wei et al. studied thepolarization dependence of the intensity of this peak inthe ssp， ppp， and sps polarizations measured with exper-imental configuration of Visible=45°， IR=57°25 TheirSFG-VS data are quantitatively very close to our datawith Config.2 as expected. Therefore， the ssp intensityof the 3693cmpeak is about 10 times of that of ppp，and the sps intensity is essentially close zero.Wei etal. realized that using the step orientational distributionfunction in Eq.10as well as other distribution functions，such as Gaussian， centered at the surface normal， can notexplain such ssp， ppp and sps intensity relationships withthe slow motion average in Eq.B On the other hand， thefast motion average centered at the interface normal， asshown in Eq with 0M =51°， can fairly well explain theobserved intensity relationships. Thus， it was concludedthat the orientation of the free OH bond of the inter-facial water molecule varies over a very broad angularrange (0M = 51°) within the vibrational relaxation timeas short as 0.5ps.25 As shown in Fig.3. Wei et al.’s treatment predictsclearly zero intensity for the sps spectra at 3693cmwith the assumption of fast orientational motion centeredat the surface normal. Using exactly the same parame-ters， our calculation of Config.2 gives the same resultsas that by Wei et al. as it should have been.25 It isclear that the simulation results in Fig.3 can fairly wellexplain the data in Config.1， 2 and 3， because all ofthem have relatively very small sps spectral intensity at3693cm. However， even though the fast orientationalmotion picture can explain the relative intensity betweenthe ssp and ppp polarization in Config.4， it is clear thatit can not explain the apparently non-zero sps intensity FIG. 3： SFG intensity of the free OH bond simulated withboth slow motion limit (solid curves) and fast motion limit(dotted curves) following the procedure and parameters asWei et al.25. 0M is the range of orientational motion of thefree OH bond. All the curves presented include the factor ofsec p， and all intensities are normalized to the ssp intensityin Config.4 with 0M =0°. The vertical lines indicate thedistribution width suggested by Wei et al. at 3693cm-1 with Config.4. As long as the orientationdistribution or orientational motion is assumed to be cen-tered to the interface normal，65 orientational distributionfunctions other than the step function in Eq.o give thesame conclusion. Since the slow motion limit is alreadynot an option，25 alternative description of the motion andorientation at the air/water interface has to be invoked. Because the air/water interface is rotationally isotropicaround the interface normal， now we assume that themolecular orientation is centered around the tilt angle0o /0， instead of the interface normal(0o=0). If theGaussian distribution function is assumed， we have in which a is the standard deviation of the angular dis-tribution. We shall show in the followings that by usingthis distribution function， the 3693cm-peak in differ-ent polarization and experimental configurations in Fig.1can be quantitatively analyzed. Because the 3693cm-1 peak belongs to the ss modeof the free O-H bond at the air water interface， it hasbeen treated with Coou symmetry. Now we calculatethe general orientational parameter c and the strengthfactor d for the ssp， sps and ppp polarizations in all fourexperimental configurations with the same parametersof the air/water interface as those used by Wei et al2 TABLE II： The general orientational parameter c and thestrength factor d for the vibrational stretching mode of free0H bond in different experimental configurations. The dvalue bear the unit Bccc of single OH bond. d-ssp c-ssp d-spsc-sps d-ppp c-ppp Config.1 0.2740.515 0.112 1 -0.1541.53 Config.2 0.256 0.515 0.118 1 -0.120) 2.05 Config.3 0.248 0.5150.118 1 -0.104 2.43 Config.4 ( 0.1760.5150.107 1 -0.0356.55 The details of the calculation of c and d can be foundelsewhere.50，51，52It is clear from Table l that the c valuesfor the ssp and sps polarizations are the same for allfour experimental configurations； whereas the c valuesof the ppp polarization differ significantly for differentexperimental configurations. ssp Configu.2 sps ppp FIG. 4： The simulated SFG intensity of vibrational stretchmode for free OH bond at different orientation angle 0 as-suminga=0°. The factor secp in Eq is also included forcomparison of SFG intensity in different experimental config-urations.All curves are normalized to the ssp intensity inConfig.4 with 0o =0°. The vertical lines indicate the orien-tation which quantitatively explains the observed SFG data. As we have demonstrated previously，49.50.51.52 the [d*r(0)]2 vs..(0 plot with a = 0 in different polarizationscan provide direct first look of the physical picture forpolarization analysis of SFG-VS data. Here we plot[d*r(0) * sec f]2 vs. 0 in Fig in order to compare datain different experimental configurations. Thus， the rel-ative intensity for the 3693cm- peak in experimentalConfig.1， 2， 3 and 4 can be used to calculate the ori-entation angle of the free O-H bond. Using the knownprocedures49.50.51，52 and parameters，25 they give the fol-lowing four values， i.e. 28.7±1.2°，32.6±0.5°，34.6±0.7° and 35.8± 1.0°， respectively.These values agree witheach other quite well. However， the value from Config.1，whose ppp and sps intensities are both very weak， is notas reliable as the other three configurations. Averagingover these values gives 0 = 33°±1°. It is clear that o=0°is not physically possible for theliquid interface. However， the apparent success of thequantitative explanation of the observed SFG spectra ofthe free O-H bond in different experimental configura-tions using a=0° indicates that the actual o value cannot be very broad. Simulation of the 3693cm-1 peak indifferent polarizations in each of the four experimentalconfigurations using the Gaussian distribution functionin Eqmconcludes that o has to be smaller than 15°to satisfy the measured 3693cm-1 peak intensities in allfour experimental configurations. o =15°is the largestdistribution width allowed by the SFG experiment datafor a Gaussian orientational distribution function. Witha = 15°， we have 0o=30°. This indeed confirms thatthe orientation of the free O-H bond is within a rela-tively narrow range (between 30° to 33°)， with a rela-tively small distribution width (o 15). Calculationwith both EqBi.e. slow average limit，and Eq.@i.e.fast average limit， gives indistinguishable results with aas small as ≤ 15° if 0o is around 30°. This is because thatwith a small distribution width， fast and slow motion av-erage should be the same according to Eqg and EqBUsing a step distribution function around 0o /0°alsogive very close orientation angle and distribution width. Thus， our conclusion of the free O-H orientation anddistribution at the air/water interface is drastically dif-ferent from the conclusion given by Wei et al.， whichconcluded that the free O-H bond orientation varies ina broad range as big as 102° and as fast as 0.5 picosec-ond， which is the relaxation time for the O-H stretch-ing vibration.25 It is clear that our conclusion is basedon the successful explanation of the observed polarizedSFG spectral intensities in different experimental config-urations， especially the relatively small but clearly non-vanishing SFG spectral intensity at 3693cm-in the spspolarization. Our conclusion explicitly supports ultrafastlibratory motions with a relatively narrow angular range.As we have known， the dynamics libratory motion of thehydrogen bonding can be as fast as 0.1 picosecond.67.68Even with such ultrafast dynamics， the air/water in-terface is nevertheless well ordered. This is consistentwith the generally well ordered picture of the air/liquidand air/liquid mixture interfaces.Recent quantitativeanalysis of data in SFG vibrational spectroscopy havesuggested that vapor/liquid interface are generally wellordered， and sometimes even with anti-parallel double-layvered structures33，56，57，58，63，69 It has been generally accepted that liquid interfacewith strong hydrogen bonding between molecules shouldbe well ordered3. Our analysis here not only confirmedthis conclusion， but also provided solid and direct exper-imental measurement of the orientation and motion atthe air/water interface. C. Polarization Analysis and Determination ofSpectral Symmetry Property Here we try to apply polarization analysis for identi-fying the symmetry property and for assignment of theSFG vibrational spectra of the air/water interface. The assignment of the SFG spectra of air/water inter-face in the range of 3000 to 3800cm- has been discussedintensively.21.22，23.24，26，27，28，37.70.71，72 Richmond recentlreviewed the current understanding of the bonding andenergetics， as well as the SFG spectra assignment，of various aqueous interfaces， including the air/waterinterface.2.28 The SFG spectral assignments heavily re-lied on band fitting of IR and Raman peak positions ofbulk water or water cluster spectra，28 as well as basedon theoretical calculations.8.9.73.74 The sharp peak atabout 3700cm- has been unanimously assigned to thefree O-H stretching vibration mode. The broad peaksaround 3250cm- and 3450cm-1undoubtedly belong tothe hydrogen bonded O-H stretching modes， but theirassignments are not as unanimous as the 3700cm1peak. The spectrum around 3250cm- was assigned toa continuum of O-H symmetric stretches(ss)，v1 of wa-ter molecules in a symmetric environment (ss-s)， and wasgenerally referred as ”ice-like”region because of its sim-ilarity in energy to O-H bonds in bulk ice.The broadband around 3450cm-1 was assigned to more weaklycorrelated hydrogen bonded stretching modes， and wascalled the "liquid-like” hydrogen-bonded region， wherewater molecules reside in a more asymmetrically bonded(as) water environment.27.28 The broad peak around3550cm-1 appeared clearly in the ppp SFG spectra hasbeen identified once and it has not been clearly assignedso far.25 Shultz et al. pointed out that these broadpeak should also include the asymmetric stretching modeof water molecules in a symmetry environment and thebending overtone.24 Richmond et al. also suggested thatthe intensity at about 3450cm-include the contributionof donor O-H bond.26 Recent progresses on SFG-VS have made it pos-sible to determine the symmetry properties of SFG-VS vibrational spectral features through comparison ofSFG spectra in different polarizations and experimen-tal configurations.50.51，52，53 The key idea of this develop-ment is from the commonsense of molecular spectroscopythat vibrational modes of molecular groups with differ-ent symmetry properties have different polarization de-pendence on the interacting optical fields.75.76 Applyingthese ideas to polarization analysis of SFG spectra hasled to a set of polarization selection rules for differentstretching vibrational modes of molecular groups withdifferent molecular symmetry properties， such as stretch-ing vibrational modes of the CH3 (C3u)， CH2 (C2u) andCH (Coou) groups.50.51.52 Many of these selection rulesare independent from molecular orientation and orienta-tional distribution. Therefore， they can be directly usedto identify symmetry property of SFG stretching vibra-tional band. These progresses make it possible to analyze FIG. 5： The simulated SFG intensity for symmetric stretching (ss) mode (left) and asymmetric stretching (as) mode (right) ofwater molecule with C2u symmetry. All curves presented include the factor of sec" p. The intensity of ss mode is normalized tothe ssp intensity in Config.4 with 0=0°. The intensity for as mode is normalized to the sps intensity in Config.4 with 0o =0°.The units between plots of the ss and as modes differ by 9.11 times according to the Becc and Baca values in the appendix. SFG vibrational spectra in situ， instead of rely only onthe assignments from Raman and IRS19 studies of the bulkphases， which can be called ex situ. Because SFG spectrausually has more features than those from IR and Ramanmeasurement， some confusions and errors in the previ-ous spectral assignments have also been clarified.50.51.52Even though SFG is naturally apolarized spectroscopyand the interfacial molecular groups are usually ordered，this idea has not been systematically explored until veryrecently.49.50，51，52 Water molecule possesses C2u symmetry. If the two O-H bond of a water molecule are asymmetrically bonded，both O-H bond has to be treated separately with Coousymmetry. This classification of the water molecule sym-metry is generally true no matter it is hydrogen-bondedor not， in cluster， in bulk or at the interface. Therefore，the symmetry property of the SFG vibrational spectrafeatures of the air/water interface can all be classifiedaccordingly. Thus， there are three kind of stretching vi-brational modes for us to deal with， namely， the sym-metric (ss) and asymmetric (as) stretching modes for C2usymmetry， and the stretching mode for Coov symmetry.It is fairly easy to distinguish these three stretching vi-brational modes from the polarization selection rules forSFG spectra of CH2 and CH groups.50.51，52 Because thebond angle of the CH2 group is slightly larger than thatof water molecule， the polarization dependence of the C2uwater molecule are slightly different. The key differenceis that even when the water molecule at the interfacerotate freely around its symmetry axis， the sps spectralintensity of its ss mode does not vanish as that for C H2.However， this fact does not make the polarization selec-tion rules different for the interfacial CH2 group and theC2u water molecule. Two of the major selection rules for the C2u group at adielectric interface are： (a) ssp intensity is always many times of that ofppp for ss mode. and (b) ppp intensityfor as mode is always several times of that ofssp. Thatis to say， if there is any peak which is stronger in thessp than ppp spectra， it can not be from the as mode.These two rules are independent from molecular orien-tation and orientational distributions at a rotationallyisotropic interface. It is clear from these selection rules， the sharp peakaround 3693cm-1 in Fig■does not belong to the C2usymmetry， because its intensit)：y in ppp polarization inConfig.1，2，3 is smaller than that in ssp polarization；while larger in Config.4. On the other hand， these all fitswell with the simulations in FigTherefore， 3693cm2-1peak is with Coov symmetry as the free O-H stretch-ing mode. Dissimilarly， both the broad peaks around3250cm-1 and 3450cm-1 in Fig are very strong only inthe ssp spectra in all experimental configurations. Theyfit well with the ss mode of the C2u symmetry， and cannot belong to the as mode of the C2u symmetry， or theCoov symmetry. It is not so easy to determine the symmetry propertyof the broad 3550cm-peak， because it appears to beburied in the high frequency tail of the broad 3450cm-peak.. It is not so straight forward to read its relativeintensity in ssp and ppp polarizations from FigHow-ever， it appears significantly bigger in sps polarization inConfig.4 than that in Config.1，2，3. Therefore， it appearsto fit with the simulations in Fig1.1In order to excludethe possibility that it may belong to a C2u mode (ss oras)， detailed simulation of the C2u modes in different po-larizations and experimental configurations is now calledupon. As describe in the appendix， the parameter c and d ofthe C2u vibrational stretching modes are calculated fordifferent polarizations and experimental configurations.Plots of [d*r(0) * sec ]2 vs. tilted angle 0 of the water molecule c axis from the interface normal using these cand d values are presented in Fig.5.These plots againconfirm that the 3693cm-1 can not belong to any C2umode， especially with the one order of magnitude increaseof this peak in ppp polarization. Clearly， the polarization dependence of the broad3550cm-peak does not fit to the ss mode of the C2usymmetry. Otherwise， according to Fig ，its p nten-sity in Config.3 has to be about one order of magnitudeweaker than that in the observed spectra. This peak cannot be the as mode of the C2u symmetry either. Accord-ing to the c and d values for the as mode of the C2usymmetry， the phases in ssp and ppp polarizations haveto be with opposite signs in all four experimental con-figurations. However， fitting of the ssp and ppp spectraindicates that in Config.4， the oscillator strengths had thesame signs for ssp and ppp polarizations， even though inCongfig. 1，2，3， the oscillator strengths of this peak dopossess opposite phases these two polarizations. This in-dicates that the 3550cm-1 peak is not C2u symmetry，and it appears to have Coou symmetry. Because in the bulk phase there is no observation offree O-H bond， and because this broad peak 3550cm-appears to be hydrogen-bonded， there is only one possi-bility that it is the other O-H bond of the interfacial wa-ter molecule which has a free O-H bond extruding awayfrom the liquid bulk phase. According to Fig.，， the CoouO-H stretching mode in sps polarization is about twiceas large in Cogfig.4 as that in the other configurations.This is fully consistent with the SFG spectra data inFigMjaand the fitting results in Table .Furthermore， inTabletthe phase of the broad 3550cm- peak is justopposite to that of the 3693cm-peak in both ssp andsps polarizations， indicating these two O-H pointing toopposite directions. The phase of the ppp polarizationof the broad 3550cm-peak changes signs with differentexperimental configuration. This is because the orienta-tional angle of the two O-H bonds are some times on thesame side of the minimum on the ppp curves in Figand sometime on the different side of the minimum， justas predicted with the experimental configuration anal-ysis. These detail features indicated the ability to un-derstand very subtle dependence of the SFG spectra onexperimental configurations and the parameters used forthe spectra calculations. Further study shall be reportedelsewhere. Further support for the assignment of the broad3550cmpeak came from the IR spectra measure-ment of the water dimer clusters， where the stretch-ing frequency for the donor O-H bond is just at about3550cm-1.66.70.2，78 This assignment is a good supportfor our assignment of the peak at 3550cm-1 in ppp spec-tra to the single hydrogen-bonded water molecule atthe interface. Furthermore， the two O-H stretching vi-brations for the methanol dimer are at 3574cm-1 and3684cm-1.9 The donor O-H stretching mode is also inthe same region of 3550cm-1. There is no observable spectra features in Fig.r.m for the as mode of the C2u water molecules， neither hydrogenbonded nor non-hydrogen bonded. According to Fig 5，for the as modes corresponding to the ss mode around3250cm-and 3450cm-， their intensities have to be atleast one order of magnitude weaker than that of theircorresponding ss modes. It is understandable that we donot observe them. Above discussion also throw doubtson the existence of interfacial water molecules with twofree O-H bonds， as suggested somewhat less convincinglyoy some recent studies.23.26 According to the po-larization selection rules and the calculation for the po-larization dependence of the C2u water molecules， no de-tectable spectral features satisfying the C2u symmetry inthe 3600cm-1 and 3800cm-1 has been observed in theSFG spectra. Here we clearly see that how polarization selectionrules， quantitative polarization and experimental config-uration analysis can help determine the symmetry prop-erty of the observed spectra features. The importanceof studying of spectral interference has been demon-strated in recent reports.23，59.80 Analysis in this work alsodemonstrated that， in order to discern spectral details，it is useful and effective to analyze the spectral interfer-ence of different spectral features through global fittingof SFG spectra in different polarizations and experimen-tal configurations， and to compare fitting results with theprediction from the calculated c and d values. This alsoindicates the usefulness of the formulation of total SFGsignal with functions of c and d parameters in Eq.M D. Molecular Structure at Air/Water Interface With the analysis of the orientation and motion， vibra-tional spectral symmetry of the water molecules at theair/water interface in above sections， we can have moreunderstanding of the molecular structure of the air/waterinterface. In Section IV.B， we have determined that the free O-H oriented around 30° away from the interface normalwith a orientational distribution narrower than g=15°，and in Section IV.C， we have identified the spectral fea-ture around 3550cm-1 of the other hydrogen bonded O-Hbond of this water molecule. If the plane of this interfa-cial water molecule is close to perpendicular to the inter-face， the orientation of the hydrogen-bonded O-H shouldpoint into the liquid phase with a orientation around135°away from the interface normal. This orientation isfully consistent with the calculation of the polarizationand experimental configuration dependence of the broad3550cm-l peak with a Coou symmetry with the observedSFG intensities， detail to be reported elsewhere. Suchorientation makes the dipole of this water molecule pointsaround 97°from the interface normal.This picture isfully consistent with conclusions in many previous experi-mental and theoretical studiesd，i5es.8 5.，183，，1230，20，81，82，83，84，85，86，87.88but certainly different from some.25 tween 3100cm-l to 3500cm-1 are determined to be sym-metric stretching modes of the C2u symmetry. Becausethe peaks are broad， and their energies is in the range ofhydrogen-bonded O-H stretching range， they can onlycome from the water molecules with two donor O-Hbonds， whose oxygen atom can accept either two， oneor zero hydrogen atom from other water molecules as hy-drogen donors. Certainly， the water molecule with theoxygen atom forming two hydrogen-bonds is tetrahedralin shape and is ”ice-like”. This is consistent with theprevious assignment of the broad 3250cm-1peak. The water molecule with no hydrogen bond for theoxygen atom is obviously with C2v symmetry. How-ever， the water molecule with only one hydrogen bondfor the oxygen atom may or may not preserve the C2usymmetry. However， if this hydrogen bond perturbationto the water structure is limited， this water moleculecan still be treated as with C2v symmetry.The lasttwo kinds of water molecules are certainly not ”ice-like”， but "liquid-like”. The two"liquid-like”speciesmay have slightly different O-H vibrational frequen-cies. However， only two apparently broad peaks in the3100cm=to 3500cmregion have been identified in theliteratures：21，22，23.2426，27，28，37，70，71，72 More studies on thepossible hydrogen-bonded species are certainly warrantedin the future. Here we confirm the conclusion by Brown et al. thatthese C2u water species all have their dipole vector pointout of the bulk liquid phase， i.e. with both hydrogenatoms point into the bulk liquid phase.23 It is clear in Ta-ble "， the signs of the ssp polarization oscillator strengthfactors of the C2u water species are all in opposite phaseto that of the free O-H peak at 3693cm-1 in all experi-mental configurations. The signs and values of the c anddparameters of the C2u and Coou in Table and Tablel，respectively， indicate that the c axis of the C2u specieshas to be in opposite direction to the c axis of the freeO-H bond at the interface. Therefore， as defined as inthe appendix， the C2u species have to have their dipolevector point out of the bulk liquid phase. The calculationof the phase of the ppp as well as sps spectral featuresare all consistent with this picture. However， because the SFG spectral intensities of theppp and sps polarizations are generally in the noise levelin the 3100cm-1 to 3500cm-1 region (FigD)， it is dif-ficult to determine the range of the orientation angle 0of these hydrogen-bonded species relative to the inter-face normal. The orientational distribution of these C2species can be quite broad， different from that for the in-terfacial water molecules with the free O-H bond. Fromour simulations， it appears to us that SFG measurementmay not be very effective to determine the orientationalangle of the C2u species at the air water interface， eventhough it can do very well with the Coov O-H bonds asshown above. However， our recent analysis of the SHGmeasurement of the neat air/water interface showed thatSHG measurement might be able to help determine theorientational angle of the C2u species， but not the Coou O-H bonds. Recent SHG results indicated that the aver-age orientation of the interfacial C2u water molecules isabout 40° to 50°from the surface normal.89 The molecular structure， orientation and dynamicsat nonpolar material/water interfaces have been stud-ied by ab initio calculation， MD simulation， or themcombined 5.6.2.8.9.10，11，12，13，90.91 It appears that some dif-ferent conclusions were drawn on the molecular orien-tation and structure of the air/water interface in dif-ferent studies.6.91.92 Nevertheless， many of these studiesconcluded that the dipole vector of the interfacial wa-ter molecules prefers lying parallel to the interface andhave one of the O-H bond protrude out of the liquidphase. The majority of the conclusions from theoreti-cal calculations agree satisfactorily with the experimen-tal analysis of ours and previous studies， but all the sim-ulation results were with significantly broader orienta-ional distributions.5.8，13，81，82，83，84.85，86，87.88 There werereports concluded that some interfacial water moleculeshave their two O-H bonds projecting into the vapor phaseand with oxygen atoms in the liquid phase.13.93.94.95 How-ever， we have not found explicit spectroscopic evidencefor such species at the air/water interface. These all in-dicate that detailed comparison of the theoretical cal-culations and the experimental analysis is certainly animportant subject in the future studies. V..CONCLUSION Detailed understanding of the air/water interface isimportant， and can be used for the general understand-ing of the liquid water structure. In this work， we pre-sented detailed analysis of the SFG vibrational spectraof the air/water interface taken in different polarizationsand experimental configurations. Polarization and ex-perimental configuration analysis have provided detailedinformation on the orientation， structure and dynamicsof the water molecules at the air/ water interface. Thesuccess of these analysis indicated the effectiveness andability of SFG-VS as a uniquely interface specific spec-troscopic probe of liquid interfaces and other molecularinterfaces. It also indicates that for the neat air waterinterface， as has been studied in the literature for someother simple air/liquid interfaces， the contribution fromthe inntterface region dominates the SFG spectra.47，96，97，98 Here are major conclusions we have reached for theair water interface. Firstly， we concluded that the mo-tion of the interfacial water molecules can only be ina limited angular range， instead rapidly varying over abroad angular range in the vibrational relaxation timesuggested previously. Secondly， because different vibra-tional modes of different molecular species at the in-terface has different symmetry properties， polarizationand symmetry analysis of the SFG-VS spectral featurescan help assignment of the SFG-VS spectra peaks todifferent interfacial species..These analysis concludedthat the narrow 3693cm-1 and broad 3550cm-1 peaks belong to Coov symmetry， while the broad 3250cm-and 3450cm-1 peaks belong to the symmetric stretchingmodes with C2u symmetry. Thus， the 3693cm-1 peak isassigned to the free OH， the 3550cm-peak is assignedto the single hydrogen bonded OH stretching mode， andthe 3250cm-1 and 3450cm-1 peaks are assigned to inter-facial water molecules as two hydrogen donors for hydro-gen bonding (with C2u symmetry)， respectively. Thirdly，analysis of the SFG-VS spectra concluded that the singlyhydrogen bonded water molecules at the air/water in-terface have their dipole vector direct almost parallel tothe interface， and is with a very narrow orientationaldistribution. The doubly hydrogen bond donor watermolecules have their dipole vector point away from theliquid phase. Finally， we did not find any observable ev-idence for interfacial water molecules with doubly freeO-H bonds at the air/water interface. Many of the conclusions in this work agree wellwith previous reports， with much more detailed under-standings. The conclusion of the narrow range mo-tion of the free O-H bond is different from the litera-ture.The explicit assignment of the broad 3550cm-peak and determination of the symmetry property of thehydrogen-bonded O-H stretching modes in the 3100cm-to 3500cm-1 region are based on firm evidences. Theseconclusions as a whole provided a detailed and generalpicture of the spectroscopy， structure and dynamics ofthe air/water interface， WS which can be used for under-standing chemical and biological problems related to theubiquitous water molecule in general. The concepts andapproaches used in the analysis in this report can be ap-plied to studying on more complex molecular interfaces. Recently， extensive efforts with SFG-VS， as well asSHG， experimental studies and theoretical simulationshave been devoted to the renewed interests on ion adsorp-tion and the Jones-Ray effect at the air/aqueous solutioninterfaces.24.64，7499.100.101.102.103.104 We suggest that de-tailed polarization and experimental configuration anal-ysis of the SFG vibrational spectra be applied to theseinterfaces. Acknowledgment. Thisworkwas supported bythe ChineseAcademy of Sciences(CAS， No.CMS-CX200305)， the Natural Science Foundation of China(NSFC， No.20425309) and the Chinese Ministry of Sci-ence and Technology (MOST， No.G1999075305). Wethank Bao-hua Wu for help derive the bond polarizabil-ity derivative model expressions. H.F.W. acknowledgesY. R. Shen for helpful discussions. Appendix： Calculation of d and c Parameters forC2u Molecule Here we present the expressions to calculate the pa-rameter c and d for water molecules with C2u symmetryusing the bond polarizability derivative model first used by Hirose et a1.105，106 The detailed re-derivation of thecomplete expressions and the effectiveness of the modelcan be found in a recent review.52 The relationship between the Raman depolarizationratio pand the bond polarizability r for a molecule groupwith C2u symmetry was：2 in which r is the H-O-H bond angle between the twoOH bonds of a water molecule.. With the Raman de-polarization ratio measured as about 0.03， the bondpolarizability r for OH bond in water molecule can bededuced to be 0.32， as used by Du et al20 The 7 hyperpolarizability tensor elements of watermolecule with C2v symmetry are as the followings. BaaclC= G-aBoH [(1+r)-(1-r)cost]cos( Where Ga = (1+cosT)/Mo +1/MH and Gb= (1-cosT)/Mo+1/MH are the inverse effective mass for thesymmetric (a1) and asymmetric (bi) normal modes， withMo and MH as the atomic mass of O and H atoms， re-spectively. wai and wbi are the vibrational frequenciesof the respective modes. on=2eacch， as defined byWei et a1.48 The water molecule are fixed in the molecu-lar coordination X'(a，b， c) with the O atom at the coor-dination center， the molecule plane in ac plane， and thebisector from the oxygen to the two hydrogen atoms sideis the c axis. For the achiral rotationally isotropic (Coou) liquid in-terface， the symmetric stretching；(ss， a1) vibrationalmodes have，52 (14) The b2 asymmetric mode are SFG inactive since theYnernolhyperpolarizability tensors Bicb and pcbb are zero. The Euler angel w can be integrated if the H-X-H planeofthe XH2 group can rotate freely around its symmetryaxis c. For water molecules with both OH bond hydrogenbonded to neighboring molecules in liquid phase， the Eu-ler angel w should not be a fixed value. Assuming a ran-dom w distribution we have the following non-vanishingtensor elements for the symmetric-stretching mode.50.52 Andthee nnon-vanishingttensorrelementsfor waterasymmetric-stretching modes are， but is very small. So the ss vibrational mode spectra inthe sps and pss polarizations should vanish as the CH2group mentioned above. However， they have to be very TABLE III： The general orientational parameterc and thestrength factor d for ss mode and as mode of water moleculewith C2u symmetry in different polarization combinations.The d values bear the unit Bccc. ss mode d-ssp c-ssp p d-sps c-sps d-ppp c-ppp Config.1 0.400 0.038 0.012 -0.146 0.174 Config. 2 0.374 0.038 0.013 -0.079 0.338 Config.3 0.3620.038 0.013 -0.046 0.589 Config.4 0.257 0.038 0.012 1 0.066 -0.378 as mode d-ssp) c-ssp d*c-sps c-sps d-ppp c-ppp Config.1 -0.154 -0.122 co 0.262 0.98 Config.2 -0.144 -0.129 0.272 0.99 Config.3 -0.139 -0.128 OO 0.277 0.99 Config.4 -0.099 1 -0.117 OO 0.250 1.01 small comparing with in ssp spectra. This is fully con-sistent with the small intensities in the sps SFG spectrafor the C2u water modes in Fig. With above deduction， anddfollowing the proce-dure in previous report， the general orientationalparameter c and strength factor d for the symmet-ric stretching (ss)) mode and asymmetric stretching(as)) mode ofWwater molecule： inndifferent polariza-tions and experimental configurations can be calculated(see Table The parameters used in the calcula-tion are ni(w)=n1(wi)=n1(w2)=1； n2(w)=n2(wi)=1.34；n2(w2)=1.18； n'(w)=n'(w1)=1.15； n'(w2)=1.09， respec-tively. These parameters are the same as the dielectricconstants used for calculation of the air/water interfaceby Wei et al.25 As we have discussed in our reports，50.51polarization analysis with the co-propagating experimen-tal geometry is insensitive to the value of the dielectricconstants of the IR frequency.2.53 Therefore， we used thesame refractive constants for the IR frequencies acrossthe whole 3100cm-1 to 3800cm-1 region， and this doesnot appear to affect our analysis.S.These c and d val-ues are used to calculate the polarization and orientationdependence of the SFG intensity， as well as the inter-ference (phase) of different spectral features in differentexperimental configurations. These calculations can sat-isfactorily explain the detailed changes of the observedspectra features， as discussed in the main text. It is to be noticed that in the above discussion we onlyused single water molecule parameters. 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