An Empirical Approach to the Bond Additivity Model in Quantitative

arXiv:physics/0602019v1 [physics.chem-ph] 3 Feb 2006 blt ooti retmt h aisbtendiffer- between ratios the polarizability estimate microscopic or ent obtain to ability ssniiet umnlyrcagsa h molecular the and at selective changes order submonolayer interface second interface. to is the sensitive of processes, is one optical as in- nonlinear SFG-VS, liquid/solid liq- because and vapor/liquid, terfaces, gas)/solid, including air(vacuum interfaces, uid/liquid, molecular ious irtoa pcrsoy(F-S xeieti early in 1987, experiment (SFG-VS) Spectroscopy Vibrational e ftemlclritrae ne study. under or- interfaces and molecular structure the been of orientational has der molecular spectra derive SFG-VS to the used of interpretation titative hmclgopi molecule. a in group chemical a oe.Teapoce mlydt banthe obtain to vibrational employed stretching approaches the The on focused modes. have efforts the of h aa oaiaiiydrvtv esr ( tensors derivative polarizability Raman the esrrto r l ae ntefloiggnrlre- general following hyperpolarizability ( order the second tensors on the based between all lationship are ratios tensor sal eoeas denote usually esr ( tensors al [email protected],Tl 61-2537 F 86-10-62555347, Tel: 10-62563167. [email protected], mail: ∗ † nEprclApoc oteBn diiiyMdli Quanti in Model Additivity Bond the to Approach Empirical An hs w uhr r iheulcontribution. equal with are authors two These uhr owo orsodnesol eadesd E- addressed. be should correspondence whom to Authors h e o uhqatttv nepeainle nthe on lies interpretation quantitative such for key The ic h rtrpr fSmFeunygeneration Frequency Sum of report first the Since 1 F-Shsbe ieyue oivsiaevar- investigate to used widely been has SFG-VS β ∂µ 2,3,4,5,6,7,8,9,10 i q ′ j k ′ odadtvt oe,btas rvdsapatclroadma practical a SFG interfaces. provides molecular the also of of but in understandings model, features new additivity all provides bond only interpret modes, not quantitatively approach as be empirical and can interface ss air/ethanol the and of elements tensor esrrto o h CH h the the for or ratios model r depolarization additivity tensor Raman bond obtained interpretat experimentally The quantitative with along interfaces. for at crucial spectrum is VS molecule a in group smerc(s tecigmdso CH of SFG-VS modes in stretching dependence (as) polarization asymmetric intensity the reproduce oe.Oeo uheape sta h smd fCH of mode as a the that the is co as examples and such such effectiveness w of systems, the agree One molecular on not uncertainties model. simplest does cast the usually has of modes certainly some as and for ss even the between intensity omk orcin otebn diiiymdl ihteco the With model. additivity bond the to fr approach corrections empirical make an as questions, to alcohol, these mo answer chain as to short order this other In while air/ethanol, interface, the from air/methanol spectra the from spectra VS ′ k /∂Q ′ .Dprmn fPyis nu omlUiest,Wh,An Wuhu, University, Normal Anhui Physics, of Department a. nweg fterto ewe ieetpolarizability different between ratios the of Knowledge o atclrvbainlmode vibrational particular a for ) .INTRODUCTION I. q u Wu Hui sal eoeas denote usually , α nttt fCeity h hns cdm fSine,Be Sciences, of Academy Chinese the Chemistry, of Institute i ′ ′ j ′ n ioemmn derivative moment dipole and ) ic h eybgnig quan- beginning, very the Since fSmFeunyGnrto irtoa Spectra Vibrational Generation Frequency Sum of a † e-a Zhang Wen-kai , .SaeKyLbrtr fMlclrRato Dynamics, Reaction Molecular of Laboratory Key State b. β 10,13,14 i ′ j ′ k 3 ′ CH , esreeet of elements tensor ofr lotall almost far, So µ 2 k ′ n Hgop.Scesul,sc ramn a quantitat can treatment such Successfully, groups. CH and , ′ fthe of ) 11,12 ∂α b † Dtd coe 9 2018) 29, October (Dated: e Gan Wei , i ′ j ′ x 86- ax: /∂Q β 3 i q ′ q j n CH and ′ th to k q ′ , b h-egCui Zhi-feng , oe i.e., mode, tem, ee( Here h rprrto ewe different if between Eq.1, ratios to proper according the Therefore, respectively. the molecule, of coordinates normal of aisbtendifferent between ratios mna RadRmnsetat orc h discrep- the correct to exper- the spectra using between Raman by ancies and model additivity IR bond imental the to approach investigated. be bond to the needs of still II. effectiveness Section model and in vibra- additivity detail limitations in (as) the discussed be stretching Therefore, shall as asymmetric symmetric SFG modes, tional and the the predict (ss) between quantitatively stretching relationship to intensity failed spectral often it in ever, mode studies. vibrational (ss) interpreta- stretching SFG-VS quantitative symmetric in the successful of tion proven theory. theory been has derivative moment it along polarizability bond analysis the bond symmetry and Raman on bond the the based II, with is Section in model detail in additivity compared be shall which eosrtdwt h sada oe fteCH the be of shall modes as approach and this ss of the with effectiveness demonstrated The groups. ular nCH in 2 mn h prahst obtain to approaches the Among nti eot esalepo ncmlt empirical complete an employ shall we report, this In rus epciey oee,terelative the However, respectively. groups, β i q dwti h odadtvt oe.This model. additivity bond the within ed ′ 3 j ω o n oaiainaayi fisSFG- its of analysis polarization and ion ′ ela ogcansratns interfaces. surfactants, chain long as well 3 to a enwdl sdt bansuch obtain to used widely been has atios mkonRmnadI pcr sused is spectra IR and Raman known om i k ru a ee enosre nSFG- in observed been never has group q cuin ae ntebn additivity bond the on based nclusions HadCH and OH ′ ′ o t plcto nSGV studies SFG-VS in application its for p j h ffcieesadlmttoso the of limitations and effectiveness the β and lmnscnb edl obtained. readily be can elements ′ ei sal eysrn o SFG-VS for strong very usually is de k i pcr o h ymti s)and (ss) symmetric the for spectra 10,13,15 ′ ′ j proaiaiiydrvtv model derivative yperpolarizability ersnstemlclrcodntssys- coordinates molecular the represents ) ′ a k rce aisbtenthe between ratios rrected rmtao nefc.Ti fact This interface. ir/methanol t ahohrwti hsmodel this within other each ith V pcr fteair/methanol the of spectra -VS n ogfiWang Hong-fei and , β ′ Q i q u rvne hn 241000; China Province, hui esreeet fachemical a of elements tensor q ′ j ′ jn,Cia 100080 China, ijing, 10,11,12,16,17,18,19,20,21,22,23,24,25,26 r h irtoa rqec,adthe and frequency, vibrational the are k ′ 3 α = CD i ′ ′ j − ′ 2 µ Hmlcls esalshow shall We molecules. OH and k ′ 2 ′ ǫ q 0 aieInterpretation tative 1 em r nw,teratios the known, are terms hvbainlmd fthe of mode vibrational th ω q µ 10,16,17,18 k ′ ∂α ∂Q ′ esr ftemolec- the of tensors i a,b ′ q j β α ′ ∗ i q i ′ ∂µ β ∂Q ′ ′ j j i ively ′ ′ ′ k j k nsm cases, some In ′ q ′ ′ em n the and terms k ′ esrratios, tensor 3 group How- (1) 2

′ ′ that with the experimentally corrected αi′j′ and µk′ ten- the local mode treatment of the CH3 and CH2 groups 28,29 sor relationships, the βi′j′k′ tensor ratios calculated from with C3v and C2v symmetries, respectively, is valid. Eq.1 can be used successfully to quantitatively inter- However, such concerns have not been a serious issue be- pret detailed polarization dependence in SFG-VS spec- cause it has been generally accepted in molecular spectra of the vapor/CH3OH and vapor/CH3CD2OH inter- troscopy textbook that ‘...in the case of C-H stretches, faces. This empirical approach overcomes the limitations the high frequency of the local vibration of the C-H bond of the current simple bond additivity model, and it also tends to uncouple that motion from that of the rest of 30 provides a practical roadmap for its application in SFG- the molecule’. In addition, slight deviation from C3v VS studies of molecular interfaces. This approach also symmetry of the CH3 group in actual molecules can be demonstrates how to use molecular IR and Raman spec- treated with small perturbations, as discussed by Hirose tra in the bulk liquid or gaseous phases for quantitative et al.16. understanding of the SFG-VS vibrational spectra of the One such approach used experimental Raman depolar- q same molecule at the interface, and vise versa. ization ratio to determine the βi′j′k′ ratios of the symmetric stretching (ss) mode of C3v and the stretching mode of C∞v groups, because both groups have only two q II. BACKGROUND independent βi′j′k′ terms for the ss mode under the local q group mode assumption, i.e., there is only one βi′j′k′ ratio 10,31 Here we briefly compare the different approaches for R = βaac/βccc = βbbc/βccc need to be determined. q obtaining βi′j′k′ tensor ratios, and discuss how and why The coordinates (a,b,c) is define as in Fig.1. This di- the bond additivity model failed to predict the SFG spec- rect Raman polarization ratio method worked because ′ ′ ′ ′ tral intensity relationship between the symmetric stretch- according to Eq.1, R = αaa/αcc = αbb/αcc for the ss ing (ss) and asymmetric stretching (as) vibrational modes mode of C3v and the stretching mode of C∞v groups, and it can be obtained from Raman experimental depolarization measurement.10 This approach has been widely used q A. Three approaches for the βi′j′k′ tensor ratios and worked successfully because it is purely empirical, and it solely depends on the reliability of the particular q Raman depolarization measurement.10,31,32,33,34 As a 3rd rank tensor, βi′j′k′ has 3 × 3 × 3 = 27 tensor elements in total. In the most ideal case, the ra- One may surmise that even though the group itself may tios between all of these tensor elements should be deter- not strictly observe the C3v or C∞v symmetry, because mined apriori. Recently, Hore et al. proposed a whole this R value is the effective value obtained from the Ra- molecule approach based on a general scheme for ab initio man spectral features with the assumption of C3v or C∞v calculation of the vibrational hyperpolarizability of any symmetry, it should dutifully reproduce back to the same IR- and Raman-active mode, regardless of the molecu- features of the Raman spectra with the same assumption of symmetry. Because this R value is strictly the same lar symmetry or complexity of the structure, and ap- q for the corresponding β ′ ′ ′ ratio according to Eq. 1, it plication of this approach to the -OSO3 headgroup of a i j k surfactant molecule at the air/water interface was also is not hard to see that with this R value and the same demonstrated.14 With the tremendous advancement of assumption of symmetry, the related features in the SFG- the capability and availability of the ab initio compu- VS spectral can be reproduced. This is why this approach has worked so well for the ss mode of the C3v and the tation methods, the advantage of this whole molecule 10,31,32,33 approach is apparent. However, because this approach stretching mode of the C∞v symmetry groups. However, this approach can not be applied to C2v is purely computational, its general accuracy can be af- q fected by the limitations of the current ab initio computa- group, which has three instead of two independent βi′j′k′ 10 tional methods in reproducing the Raman and IR spectra terms for its ss mode. Furthermore, this approach does q intensities, as well as in dealing with the problem of mode not deal with the ratio between the βi′j′k′ of symmetric couplings, such as Fermi resonances. Nevertheless, this and asymmetric stretching modes of the C3v molecular attempt does provide an promising alternative solution group. Therefore, it can not be used to address the rel- to the limitations of other existing approaches. ative SFG intensity of the ss and as modes of the C3v Unlike the whole molecule approach mentioned above, molecular group. these other approaches all employed the local mode assumption of the particular molecular group. The bond additivity model even further employed the local mode An useful alternative method in the SFG-VS litera- assumption of the particular bonds. With the knowl- ture is the widely used bond additivity model, which q edge of the symmetry of the molecular groups, the num- can give β ′ ′ ′ ratios of the C3 , C2 , C∞ and other q i j k v v v ber of non-zero βi′j′k′ tensor elements can be signif- symmetry groups, including both of their ss and as icantly reduced; accordingly, the number of non-zero modes.10,11,12,16,17,18,19,20,21,22,23,24,25,26 The bond addi- ′ αi′j′ elements of the symmetry group can also be sig- tivity model can also be called the bond polarizability 15,27 14 nificantly reduced. For example, the CH3 and CH2 derivative model. As Hore et al. pointed out recently, groups can be treated with C3v and C2v symmetries, re- the key to the bond additivity model is that through q spectively. Thus, their non-zero βi′j′ k′ elements for the symmetry analysis the polarizability tensors of the indi- stretching vibrational modes are reduced to 11 and 7, vidual bond stretches (C∞v) are coupled to produce the respectively.15,27 There have been discussions on whether normal mode coordinate of a particular molecular group 3

c called the zero-order bond moment theory, as used in the bond additivity model in SFG-VS, has not been success- 36,41,42,43,44 H1 H2 ful in interpretation of IR intensities. This is because the simple bond moment theory essentially ne- H3 glects all coupling effect between the single bonds even a within the same molecular group. Therefore, modified C bond moment theory which includes such and other coupling effects has been proposed,36 but such modified theory becomes complicated by introduced many unknown b parameters in order to address the coupling terms. In short, these facts certainly limit the effectiveness of the FIG. 1: Molecule-fixed(abc) axis for C3v symmetry. Illus- bond additivity model for quantitative interpretation of trated with the CH3 group. the SFG-VS spectra. However, with all these problems, two of the impor- 16,17 (C3v, C2v, etc.). There are two ways to determine tant and successful features of the bond additivity model the polarizability tensors ratios of the individual bond should not be overlooked. Firstly, the bond additivity stretches, i.e. r = βξξζ /βζζζ = βηηζ /βζζζ , where (ξηζ) is model rooted deeply into the concept of molecular sym- the single bond fixed coordinates with ζ as the primary metry analysis, which is the basis for the polarization axis of the single bond. One way is through theoreti- selection rules and has provided clear physical picture cal (ab initio) calculation of the Raman tensors of the for understanding of the polarization and orientational 10,25,26 single bond,35 and the other way is through experimen- dependence in SFG-VS, as well as all other spec- 45 tally measured Raman depolarization ratio, or Raman troscopic techniques. Secondly, the direct Raman polar- intensity ratio between the ss and as modes,10,16,17 on ization ratio method has been quantitatively successful in 10,31,32,33,34 the basis that the single bond posses C∞v symmetry and many of the SFG-VS applications. Since a full ′ ′ ′ ′ computational solution for quantitative interpretation of that r = αξξ/αζζ = αηη/αζζ for a single bond according to Eq.1. the SFG-VS spectra is yet not within our reach as men- The detailed formula and the effectiveness, as well as tioned above, it can certainly be beneficial to extend the some limitations, of the bond polarizability model was bond additivity model if we can use the core concept for critically reviewed recently.10 So far, this model have its successes, and try to directly address the limitations been used for study of H2O molecule at the air/water in the consisting Raman and IR theories. 19,20,21,22 18,24,25,26 interfaces, -CH2- groups and occasion- 11,12,17,23 ally -CH3 groups at various molecular interfaces. The unique success of the bond additivity model has B. Failure of the bond additivity model been on H2O molecules and -CH2- groups to which direct Raman polarization ratio method can not be applied as The failure of the bond additivity model came to mentioned above. In terms of -CH3 or other C3v groups, the bond additivity model is essentially the same thing as our attention in our recent study of SFG-VS spec- the direct raman depolarization ratio method for the ss tra of vapor/alcohol interfaces of C1-C8 alcohols in the CH stretching vibration region between 2800 to mode. Another unique advantage of the bond additivity −1 10,26 q 3000cm . The fact has been long known that there model is that it can give the βi′j′k′ ratio between the ss 10,26 has been no observation of the CH3-as peak, which and as modes for the C3v group. However, some in- should be around 2970cm−1, in the SFG-VS spectra of trinsic weakness of the bond additivity model has limited the vapor/methanol interface;12,26,46,47,48,49,50,51,52 while its effectiveness on interpretation of the relative intensity for longer chain alcohols (C2 − C8 alcohols), the CH3- of the ss and as modes in actual SFG-VS spectra. This as peak is apparently very strong in their SFG-VS is discussed in detail as in the followings. spectra.26,48,52 Even though the bond additivity model can seemingly explain why the longer chain alcohols can have stronger CH3-as peak in the SFG-VS spectra of their vapor/liquid interfaces than that of the va- 10 por/methanol interface, it also predict that the CH3-as The bond additivity model uses the Raman bond po- mode of methanol should be observable in the sps SFG- ′ larizability derivative tensors αi′j′ and the bond moment VS spectrum. ′ ′ ′ ′ derivative tensors µk′ to calculate βi j k tensors accord- Fig.2 presents the SFG-VS spectra and the calculated 10,17 ing to Eq.1. In Raman and IR spectroscopy stud- polarization dependence against the CH3 orientational ies, the bond polarizability theory and the bond moment angle θ from the interface normal using the bond addi- theory have been used to interpret the IR and Raman tivity model in the ssp, ppp, and sps polarizations for spectral intensity.36 As we have known, even though the two different experimental configurations. The SFG-VS Raman bond polarizability derivative model have worked intensity is proportional to sec2(β)d2R(θ), and the de- well with the intensities of the ss modes, it has not been tail of calculation has been presented previously.10,26 It very effective on the relative intensity between the ss is to be noted that calculation of the ss mode inten- 36,37,38,39,40 2 2 and as modes of the same molecular group. sity uses βccc as unit, and the as mode uses βaca as Furthermore, the simple bond moment hypothesis, also unit. It is clear that for both experimental configura- 4

1.0 0.5 -6 14x10 o o Vis 62 IR 53 o o

sspss Vis 62 IR 53 sec 12 ssp 0.8 0.4 ) 2 2 ( β ccc CH3-ss ppp )d 10 β

pppas 2

sps 0.6 0.3 R(

8 θ )-as ( )-ss ( θ 6 0.4 0.2 R(

2 sspas

)d spsas β

4 β aca (

2 spsss

0.2 0.1 2 ) SFG Intensity (a.u.) 2 sec pppss 0 0.0 0.0 0.5 -6 14x10 o o 1.0 Vis 37 IR 51 o o

Vis 37 IR 51 sec

12 ) 0.4 2 0.8 2 ( ccc CH3-Fermi β β 10 )d

sspss 2 0.3 R( CH3-Combi 0.6 8 θ )-as ( )-ss ( θ

6 R( pppas 2 0.4 pppss 0.2 )d β

β spsas ( 4 aca 2 sspas 2 SFG Intensity (a.u.) 0.2 0.1 ) 2 sec spsss 0.0 0.0

2750 2800 2850 2900 2950 3000 0 20 40 60 80 -1 Wavenumber (cm ) Orientation angle(θ)

FIG. 2: SFG spectra of the air/methanol interface in two different experimental configurations with different incident angles. The β in sec2(β) is the angle between the SFG signal beam and the surface normal. It is determined by the visible and IR 2 2 2 2 incident angles. The SFG intensity is proportional to sec (β)d R(θ) in units of βccc or βaca for the ss and as modes, respectively. The solid lines in the two figure on the left are fittings with Lorentzian lineshapes.

3.0 0.30 -6 20x10 o o CH3CD2OH Vis 62 IR 53 ssp sspss 2.5 0.25 ppp pppas

sps sec ) 2 15 2 (

2.0 0.20 b ccc )d b 2 CH3-as R( q )-as ( )-ss ( 1.5 0.15 10 q spsas R( sspas 2 )d CH3-Fermi 1.0 0.10 b b pppss aca CH3-ss CH3-Combi ( SFG Intensity (a.u.) 2 2 )

sec spsss 5 0.5 0.05

0.0 0.00 0

2800 2850 2900 2950 3000 0 20 40 60 80 -1 Wavenumber (cm ) Orientation angle (q)

◦ FIG. 3: SFG-VS spectra of the vapor/CH3CD2OH interface in ssp, ppp and sps polarizations with incident angles of Vis=62 ◦ and IR=53 in the SFG experiment. Because the CH2 group is deuterated, therefore, all spectral features belong to the CH3 group. The solid lines are ﬁttings with Lorentzian lineshapes.

−1 tions, CH3-as mode around 2970cm can not be ob- cated in Fig.2. This fact certainly suggests the effec- served in the SFG-VS spectra. This tells us that the tiveness of the bond additivity model calculations for missing of the CH3-as mode peak is not something ac- the CH3-ss mode of methanol molecule. Accordingly, cidental. With the bond additivity model, methanol has if the bond additivity calculation is also effective for the βccc/βaca = 1.1 for its CH3 group, as calculated in Sec- as mode, the CH3-as mode intensity in the sps polar- tion III. Therefore, from the calculation results in Fig.2, ization should be more than 1/3 of that of the CH3-ss CH3-as should be experimentally observable in at least mode in the ssp polarization of the first experimental one of the three polarizations in comparison to the inten- configuration. However, no such CH3-as mode has been 12,26,46,47,48,49,50,51,52 sities of the CH3-ss mode. Recent measurements have ever observed. This clearly demon- determined that for vapor/methanol interface the CH3 strates the failure of the bond additivity model in dealing 10,50,51,53 orientation is close to the interface normal. With with the CH3-as mode, even though the intensities of the this CH3 group orientation, the bond additivity model CH3-ss and CH3-ss-Fermi modes can be quantitatively predicted that the CH3-ss mode in the ppp polarization explained as in Fig.2. can be observed in the second experimental configura- ◦ ◦ In comparison, Fig.3 presents the SFG-VS spec- tion (VIS=37 and IR=51 ), and this spectral feature tra and the bond additivity model calculation of was indeed observed with the expected intensity as indi- vapor/CH3CD2OH interface with the incident angles 5

◦ ◦ Vis=62 and IR=53 in the SFG-VS experiment. It is vapor/CH3OH and vapor/CH3CD2OH interfaces. −1 clear that the CH3-as mode around 2970cm in the For the stretching vibrational modes of a molecular ppp polarization is even much bigger than the CH3-ss group with C3v symmetry, there are a single symmet- −1 mode around 2870cm in the ssp polarization. This ric mode (A1) and a doubly degenerated asymmetric is consistent with the bond additivity model value of mode (E), with the following relationships between the ′ ′ ′ ′ ′ βccc/βaca = 0.30 for the CH3 group of CH3CD2OH βi j k , αi′j′ and µk′ tensors when there is no electronic molecule, as calculated in Section IV. Then, why the resonances.10,54,55 bond additivity model works for the CH3-as mode of For the A1 (symmetric) mode: CH3CD2OH but not that of CH3OH? Even though the bond additivity model and the direct SFG-VS βaac = βbbc = Rβccc (2) Raman polarization ratio method have been apparent ′ ′ ′ Raman: αaa = αbb = Rαcc (3) success in quantitative interpretation of the polarization ′ IR: µc (4) dependence and orientational analysis of the CH3 groups from the SFG-VS spectra of its CH -ss mode,10,50,51 3 For the E (asymmetric) mode: their inability and discrepancy to address the as mode intensities of the CH3 group in CH3CD2OH and CH3OH SFG-VS βaca = βbcb = βcaa = βcbb molecules are certainly unpleasant, and put uncertain- − − − ties to these models. However, as discussed above, the βaaa = βbba = βabb = βbab (5) ′ ′ ′ ′ bond additivity model is still the most useful model in Raman: αac = αca = αbc = αcb q ′ ′ ′ ′ obtaining the βi′j′ k′ tensor ratios. We realized that the αaa = −αbb = −αab = −αba (6) eﬀectiveness of these models on the ss mode of the C3v ′ ′ IR: µa = µb (7) and the stretching mode of the C∞v symmetry lies on the fact that their R values are empirically obtained. On the ′ ′ It is to be noted that the αaa and αbb tensors of the other hand, the failures of these models also lie on the ′ ′ symmetric mode are diﬀerent from the αaa and αbb ten- fact that those other ratios were not set on the same em- sors of the asymmetric mode. pirical basis. In order to address these problems and to For a rotationally isotropic interface, the four asym- have better understanding of the bond additivity model, metric βi′j′ k′ terms, namely βaaa = −βbba = −βabb = we propose an complete empirical approach in order to −βbab, are dumb tensors, which would not appear in the correct the failures of the bond additivity model. This non-zero elements of the 3rd rank macroscopic suscepti- 10 approach includes the full empirical treatment of the ss bility χijk tensors. Therefore, if R = βaac/βccc is known vs. as ratios of the raman and IR tensors, in addition to from the direct Raman depolarization ratio method, in the successful empirical treatment of the ss mode. addition, only the ratio βccc/βaca need to be obtained for 10,17 all useful βi′j′k′ tensor ratios. According to Eq.1, one has, III. BOND ADDITIVITY MODEL AND ITS EMPIRICAL CORRECTION ′ ′ βccc ωasym αcc µc = ′ ′ (8) In a recently review on quantitative treatment on SFG- βaca ωsym αac µa VS from molecular interfaces, we discussed related issues on the bond additivity model in detail.10 We pointed out Therefore, with the frequencies of the symmetric 16,17 that the original expression given by Hirose et al. and asymmetric modes (ωsym and ωasym, respectively) ′ ′ needed to be corrected for some minor errors, and the known, only the Raman tensor ratio αcc/αac and the IR ′ ′ correct expressions for the C3v,C2v and C∞v groups were tensor ratio µc/µa need to be known to make the calcu- 10 presented. We also presented some good examples to lation of βccc/βaca. ′ ′ demonstrate the validity of the bond additivity model for The IR tensor ratio µc/µa can be directly obtained quantitative analysis of SFG-VS spectra, especially for from the IR intensity ratio of the symmetric and asym- interpretation of the ss modes of the C3v, C2v and C∞v metric modes. However, in the simple bond addi- ′ ′ groups, and we also pointed out that further detailed tivity model, the µc/µa ratio was not experimentally examinations of this model were still needed.10 corrected.10,17 Therefore, here we shall try to keep the successful part of the bond additivity model, which is to use the exper- IR ′ 2 ′ 2 imental Raman depolarization ratio of the ss mode to Isym µc µc IR = ′ ′ = ′ (9) obtain the single bond Raman polarizability derivative Iasym µa + µb 2µa tensor ratio r, and try to derive the expressions for the ′ ′ Raman and IR intensity expressions for the ss and as In order to get R and αcc/αac, the following Raman mode of the C3v group, in order to obtain the correc- polarizability theory need to be invoked. ′ ′ ′ ′ When vertically linear polarized light is used for the tions factors for the αi j ratios and µk′ ratios between the ss and as modes from the experimental Raman and Raman experiment, the following relationships for the IR spectra, respectively. We hope this approach can pro- Raman depolarization ratio and Raman intensity ratio q duce the corrected β ′ ′ ′ tensor ratios between the ss and in the parallel polarization to the individual Raman po- i j k ′ as modes, and can be used to quantitatively address the larizability derivative tensors (αi′j′ ) are generally valid above discussed problems in the SFG-VS spectra of the for all molecular groups.54,56,57 6

2 ∂αζζ 3γ a0 = ( )0 ρ = (10) ∂∆ζ 45α2 +4γ2 Raman 2 4 2 ∂µζ I α + γ m0 = ( )0 sym = sym 45 sym (11) ∂∆ζ IRaman 2 4 2 asym αasym + 45 γasym 1+2cos τ 1 GA1 = + MA MB with the deﬁnition of 1 − cos τ 1 GE = + MA MB

1 ′ ′ ′ in which τ is the tetrahedral angle between two single α = (αaa + αbb + αcc) (12) 3 A-B bonds, MA and MB are the atomic masses of the 2 1 ′ ′ 2 ′ ′ 2 atom A and B of the AB3 with C3v symmetry. Therefore, γ = [(αaa − αbb) + (αbb − αcc) 2 GA1 and GE are the inverse reduced masses of the A1 2 2 2 2 +(α′ − α′ ) + 6(α′ + α′ + α′ )] (13) and E normal modes, respectively, with the normal mode cc aa ab bc ca coordinates defined as,16,17,58 The molecular coordinates are defined as in Fig.1. If natural light is used for the Raman spectra measurement, 1/2 QA1 = (∆r1 + ∆r2 + ∆r3)/(3GA1 ) the factor 4/45 used in Eq.11 should be 7/45, and the de- − − 1/2 polarization ratio becomes ρ =6γ2/(45α2 +7γ2).54,56,57 QE,a = (2∆r1 ∆r2 ∆r3)/(6GE) 1/2 In this case, if one puts Eq.3 into Eq.10, because there QE,b = (∆r2 − ∆r3)/(2GE) are only three non-zero symmetric tensors, a unique R in which ∆ri is the bond displacement vector along the value (generally 1 ≤ R ≤ 4forC3v groups, and 0 ≤ R ≤ 1 direction of the ith bond, with the C3v coordinates de- for C∞v groups) can be obtained from the symmetric fined as in Fig.1. Then according to the bond additivity tensors through the experimental value of the Raman ′ 10,13 model, we have the following expressions for the α ′ ′ and depolarization ratio ρ of the C3 symmetric mode. i j v ′ µk′ tensors of the C3v group. For the A1 (symmetric) mode: 2 3γsym ′ ′ αaa = αbb ρsym = 2 2 45αsym +4γsym 1 1 2 2 = a0[(1 + r) − (1 − r)cos τ](3G 1 ) 3 2 A = 2 (14) 1 4 + 5[(1 + 2R)/(R − 1)] ′ 2 2 2 αcc = a0(cos τ + r sin τ)(3GA1 ) 1 ′ − 2 Here the Eq.14 for obtaining R is the so called direct µc = m0 cos τ(3GA1 ) Raman depolarization ratio method because R directly For the E (asymmetric) mode: satisfies Eq.2 according to Eq.1.10,31 ′ ′ ′ However, since the raman depolarization ratio for the αac = αca = αbc asymmetric mode ρasym = 3/4 is always a constant, no 1 ′ 1 2 knowledge of the asymmetric tensors can be learned from = αcb = a0(r − 1) sin τ cos τ(6GE ) 2 the direct Raman depolarization ratio method. There- α′ = −α′ = −α′ fore, one now have to employ the bond additivity model aa bb ab 1 1 to make such connection through Eq.11. ′ 2 2 = −αba = a0(1 − r) sin τ(6GE ) In the bond additivity model, the single bond prop- 4 1 1 erty is used to calculate the property of the whole group. ′ ′ 2 µa = µb = m0 sin τ(6GE ) Following Hirose et al.’s formulation,17,58 the Raman po- 2 larizability derivative of the single bond can be related It is to be noted that the above expressions is de- to the following three non-zero tensor elements with the rived with the assumption that the C3v group closely following relationship, assumes a normal tetrahedral structure. For most the CH3 groups under our investigation, this is generally a good assumption. In order to simplify calculations, here ∂αξξ ∂αηη ∂αζζ we use cos τ = −1/3, which is exact for the normal tetra- = = r (15) ◦ ∂∆ζ ∂∆ζ ∂∆ζ hedral angle τ ≈ 109.5 . Therefore, with MC = 12 and

MH = 1, we have GA1 /GE = 37/40. in which ζ is the primary axis of the single bond coordi- In order to make the bond additivity model work, the nates, and the stretching vibration displacement ∆ζ from single bond value of r need to be determined. The r value its equilibrium position is along the single bond. Gener- can be determined from computations,35 or from Raman 10 10,16,17,58 ′ ally, for the single bond, 0 ≤ r ≤ 1. This is to say, the experimental measurement. . With above αi′j′ single bond is treated with C∞v symmetry. In the bond expressions put into Eq.10 and Eq.11, we have the fol- additivity model,10,16,17,58 we deﬁne the following, lowing two expressions which can each produce an r value 7 from the experimental Raman depolarization ratio or the would generally fail because this simple bond moment Raman intensity ratio between the ss and the as modes, model has not included the coupling between the single respectively. For the Raman measurement with a linearly bonds. Therefore, in order to get the correct βccc/βaca ′ ′ polarized light, we have, ratio according to Eq.8, the µc/µa ratio also needs to be corrected with the measured IR spectra using Eq.9. ′ ′ ′ ′ 0.75 We hope by putting the corrected αcc/αac and µc/µa ρ = (16) ratios back into Eq.8, the corrected β /β ratio thus sym − 2 ccc aca 1+11.25[(1 + 2r)/(1 r)] obtained can be used to address the failures of the bond Raman 2 2 Isym 45(1+ 2r) + 4(1 − r) GA1 additivity model discussed in Section IIB. If we can do Raman = 2 (17) Iasym 32(1 − r) GE so, it shall put the uncertain parameters used in the bond additivity model on a solid empirical basis. In the next 2 − 2 ′ 2 45(1 + 2r) + 4(1 r) αcc section we shall used the well established IR and Ra- = 2 ′ (18) 8(1+8r) αac man data of liquid CH3OH and CH3CD2OH molecules in the previous literatures to demonstrate how this em- with the following relationships, pirical approach to the bond additivity model works. The terms and expressions for the C2v groups are listed in the 1 ′ 2 Appendix. αcc 1+8r GA1 ′ = (19) αac 2 − 2r GE 1 ′ 2 µc 1 GA1 IV. RESULTS AND DISCUSSION ′ = (20) µa 2 GE β ω 1+8r G A. Bond Additivity Model Results ccc = asym A1 (21) βaca ωsym 4(1 − r) GE The Raman depolarization ratio for the ss mode of Some of the Raman spectra were measured with natu- the CH3 group in CH3OH and CH3CH2OH molecules rally polarized light, then the Eq.17 and Eq.18 becomes, in bulk liquid at the room temperature measured with polarized laser light are ρ = 0.014 and ρ = 0.053, 56 Raman 2 2 respectively. Photoacoustic stimulated Raman mea- Isym 45(1+ 2r) + 7(1 − r) GA1 = (22) surement of CH3CH2OH, CH3CD2OH and CD3CH2OH Raman − 2 Iasym 56(1 r) GE in the gas phase gave ρ = 0.060 ± 0.015 for all three 59 45(1 + 2r)2 + 7(1 − r)2 α′ 2 molecules. These values are in good agreement with = cc (23) 14(1 + 8r)2 α′ each other. Because the previous Raman experiment ac claimed to have higher accuracy (error of ρ was claimed One would naturally expect that Eq.16 and Eq.17 (or to be 0.0007), and was measured with bulk liquids, here Eq.22) would end up with the same r value from Ra- we adopt the value ρ = 0.053 for CH3CD2OH in this man experiment measurement on the same molecule.17 work. From Eq.14 and Eq.16, the R and r values for the However, experimental data have shown that it has gen- CH3 group in CH3OH and CH3CH2OH molecules can be erally not been the case. Therefore, in the bond addi- directly calculated. tivity model, Eq.16 and Eq.17 can not be used at the Therefore, for CH3OH, R = 1.7 and r = 0.28; while same time. This fact has not been clearly recognized for CH3CH2OH, R =3.4, and r =0.026. since the ﬁrst derivation of the bond additivity model by Using the two r values, the following values can be Hirose et al.17 Now the question is whether one of them obtained from the uncorrected bond additivity model for can be used, or both are not good at all. As we have the CH3 group in CH3OH and CH3CH2OH, using the known that the direct Raman polarization ratio method three equations Eq.19-Eq.21. −1 using Eq.14 involves only with the symmetric Raman For CH3OH, with r = 0.28, ωsym = 2836cm and −1 tensors of the group, and it has been widely and success- ωasym = 2987cm , we have, fully employed.10,31,32,33,34 As has been shown before, the bond additivity model with the single bond r value from ′ ′ Eq.16 is equivalent to direct raman depolarization ratio αcc µc βccc 10 ′ =2.2 ; ′ =0.48 ; =1.1 (24) method with Eq.14. Therefore, it is reasonable to stick αac µa βaca with the r value obtained from Eq.16, and then try to ′ ′ −1 experimentally correct the αcc/αac ratio with Eq.18 by For CH3CD2OH, with r = 0.026, ωsym = 2870cm −1 using the r value obtained from Eq.16. We surmise such and ωasym = 2976cm , we have, correction directly from the Raman depolarization ratio measurement and the Raman intensity measurement can ′ ′ correct the value for the asymmetric tensors, and in the αcc µc βccc ′ =0.60 ; ′ =0.48 ; =0.30 (25) meantime keep the relationship between the symmetric αac µa βaca tensors intact. Here Eq.20 from the bond additivity model can be di- These values were the same as calculated in the liter- rectly tested with IR spectral measurement using Eq.9. ature, and were used to make the calculations for Fig.2 As we have know from the discussion in Section II, it and Fig.3.10 8

1.2 TABLE I: Integrated Raman56 and IR60 band intensities of Raman 1.0 liquid methanol at 298K. All values are normalized to the CH3-as strongest peak in the Raman and IR spectra, respectively. 0.8 Band ν(cm−1) FWHH(cm−1) Intensity 0.6 CH3-ss Raman CH3-ss 2836 20 1.00 CH3-Combi 2885 39 0.16 0.4 3 CH -Combi 2918 28 0.26 0.2 Raman Intensity (a.u) CH3-Fermi 2944 24 0.84 0.0 CH3-as 2987 40 0.20 IR IR CH3-ss 2833 24 0.60 1.0 CH3-Combi 2872 39 0.18 0.8 CH3-Combi 2911 26 0.98 0.6 CH3-Fermi 2946 37 1.00

CH3-as 2983 47 0.70 0.4

IR Intensity (a.u) 0.2

B. Empirical Corrections of Bond Additivity 0.0 Model Results 2800 2850 2900 -1 2950 3000 Wavenumber (cm ) ′ ′ In order to make empirical corrections of the αcc/αaa ′ ′ and µc/µa ratios in the bond additivity model, the rel- FIG. 4: Raman and IR spectra (solid dots, normalized to the ative intensities of the CH3-ss and CH3-as peaks in the strongest peak in each spectra) of pure liquid CH3CD2OH at 61 Raman and IR spectra need to be known. 300K, redrawn from previous literature. The dashed lines are fittings with Gaussian lineshape. The base lines are determined by the best Gaussian group fittings. The area of each Gaussian curve is integrated and normalized as the cor- A detailed band fitting of the Raman spectra from a responding peak intensity listed in Table II. polarized Raman measurement of liquid CH3OH at 298K were reported by Griffiths et al. in 1972.56 The integrated band intensity of the four C-H stretching bands thus ob- TABLE II: Integrated Raman and IR band intensities of liq- tained are listed in Table I. The intensity ratio between uid CH3CD2OH at 300K. All values are normalized to the strongest peak in the Raman and IR spectra, respectively. the CH3-ss and CH3-as is thus 1.00/0.20=5.00, which is ′ ′ −1 used in Eq.18 to calculate the corresponding αcc/αac. Band ν(cm ) FWHH Intensity A detailed band fitting of the IR spectra of liquid Raman CH3-ss 2871 22 0.36 CH3OH at 298K were reported by Bertie et al. in CH3-Combi 2899 30 0.25 60 1997. The integrated band intensity of the C-H stretch- CH3-Fermi 2930 17 1.00 ing bands are also listed in Table I. It is to be noted that CH3-as 2972 22 0.67 the Raman and IR band positions and band widths (Ta- ble I) are in very good agreement with each other. The IR CH3-ss 2875 20 0.20 CH3-Combi 2900 27 0.29 intensity ratio between the CH3-ss and CH3-as is thus 0.60/0.70=0.86, which is used in Eq.9 to calculate the CH3-Fermi 2932 27 0.69 ′ ′ corresponding µc/µa value. CH3-as 2971 22 1.00 The Raman and IR spectra of 12 kind of deuterated ethanol molecule were systematically studies by Perchard and Josien in 1968.61 Fig.4 shows the Raman (mea- Now the empirical correction for the bond additivity sured with polarized light) and IR spectra of pure liq- model can be readily made with Eq.9, Eq.21, and Eq.8 uid CH3CD2OH at 300K reproduced from Perchard and as the following. −1 Josien’s paper. With the reproduced spectra, we per- For CH3OH, with r = 0.28, ωsym = 2836cm and −1 formed band fitting and listed the integrated band in- ωasym = 2987cm , we have, tensities in Table II. therefore, the intensity ratios between the CH3-ss and CH3-as of the Raman and IR spec- ′ ′ tra are 0.36/0.67=0.54, and 0.20/1.00=0.20, respectively. αcc µc βccc ′ =1.9 ; ′ =1.9 ; =3.8 (26) Comparing with the values for CH3OH above, it is clear αac µa βaca that the relative CH3-as band Raman and IR intensities −1 are many times stronger for CH3CD2OH than those for For CH3CD2OH, with r = 0.026, ωsym = 2871cm −1 CH3OH. and ωasym = 2972cm , we have,

′ ′ αcc µc βccc ′ =0.34 ; ′ =0.89 ; =0.31 (27) αac µa βaca 9

the calculated intensities in Fig.2 with Vis=62◦ and TABLE III: Comparison of the bond additivity model and IR=53◦ experimental configuration, Isps /Issp ≈ corrected bond additivity model values. CH3−as CH3−ss 1/2.5 with the uncorrected bond additivity model value CH3OH CH3CD2OH βccc/βaca = 1.1, and with the known orientation an- BAM Corrected BAM Corrected gle θ ≈ 0 from previous studies.50,51,53 Now, with the βaac sps ssp R = βccc 1.7 1.7 3.4 3.4 corrected value βccc/βaca = 3.8, the ICH3−as/ICH3−ss r 0.28 0.28 0.026 0.026 ratio should be (3.8/1.1)2 = 11.9 times smaller, i.e., sps ssp βccc ≈ 1.1 3.8 0.30 0.31 ICH3−as/ICH3−ss 1/30. It is easy to see that such βaca sps ssp ′ ′ a small I comparing to I is below the noise αcc/αac 2.2 1.9 0.60 0.34 CH3−as CH3−ss ′ ′ level and its spectral feature can not be observed in the µ /µ 0.48 1.9 0.48 0.89 c a SFG-VS spectra. Convincingly, using above approach, inspection of the IR and Raman spectra of liquid ace- C. Discussion tone and acetonitrile also indicated that the CH3-as vs. CH3-ss intensity ratio values are significantly smaller than the values calculated from the simple bond addi- Table III summarizes the values of R, r, βccc/βaca, ′ ′ ′ ′ tivity model for their SFG-VS spectra from vapor/liquid αcc/αac, µc/µa in the uncorrected and corrected bond interfaces.23,62,63,64 additivity model. It is easy to see that the biggest correc- It is interesting to see that the correction for the tion is the µ′ /µ′ value for CH OH. According to the sim- c a 3 β /β value is very small for CH CD OH, even ple bond moment theory, µ′ /µ′ =0.48 for both CH OH ccc aca 3 2 c a 3 though the corrections for α′ /α′ and µ′ /µ′ are not so and CH CD OH, i.e. the IR intensity for the CH -as cc ac c a 3 2 3 small separately. These two corrections happened to can- mode should be about 2(1/0.48)2 = 8.7 times of that cel each other when used to calculate β /β with Eq.8. for the CH -ss mode. However, experimentally they are ccc aca 3 Using this ratio, the SFG-VS spectra of CH CD OH can about 1.2 and 5 for CH OH and CH CD OH, respec- 3 2 3 3 2 be well interpreted just as with the uncorrected bond tively. Such big difference between these two molecules additivity model.10 However, it is to be noted that the can be attributed to the fact that there is much stronger success of the former is with firm empirical basis, and influence on the CH group stretching vibrations by the 3 the success of the latter is nevertheless coincidental. We O-H group in CH OH than in CH CD OH. 3 3 2 have performed measurements on the SFG-VS spectra in It is interesting to see that r =0.026 for CH CD OH, 3 2 different incident angles for the vapor/ethanol interface, while r = 0.28 for CH3OH. According to the defini- and analysis show that with the β ′ ′ ′ tensor ratios ob- tion of r in Eq.15, r = 0 means that each bond can i j k tained here, all spectra as well as the spectra interference be pictured as a perfect rod. Therefore, the value of r effects, can be quantitatively interpreted, and detailed may be used to correlate the coupling between the dif- molecular orientation analysis is also possible.34 ferent bonds in the same group, and the perturbation of the bonds in the group by the connecting molecu- According to above discussion, the success of the corrected bond additivity model is evident. This success lar groups. The CH3 group in CH3OH is likely to be much more strongly perturbed by the directly connect- indicates that the failure of the bond additivity model discussed in Section IIB can be quantitatively corrected ing O-H group, than the CH3 group in CH3CD2OH by with the empirical approach as demonstrated here. How- the directly connecting CD2 and the indirectly connect- ever, we have to caution that this success is purely empir- ing O-H groups. The same effect as in CH3OH can also be expected for acetone and acetonitrile molecules, ical, and may not be as general as it seems before further examinations being made. Even though it is undoubtedly where the CH3 groups are directly connected to the - C=O and -C≡N groups, respectively. Similarly, the an advancement from the simple bond additivity model, there are reasons this empirical approach may fail. longer chain alcohols should be similar to CH3CD2OH. Therefore, it is not coincidental that the asymmetric Because the correction for the bond additivity model C-H stretching mode in the SFG-VS spectra from the used the Raman and IR spectra in the liquid phase, the vapor/acetone and vapor/acetonitrile interfaces has not success of this correction undoubtedly indicates that the been observed,23,62,63,64 while for the longer chain alco- CH3 group of these molecules under studying are not sig- hols it has been experimentally observed.5,26,48 nificantly different as in the liquid bulk phases or at the interfaces. This may only be true for the generally ‘rigid’ C-H stretching vibrations which is usually not greatly perturbed and the assumption of local motion is generally Raman Raman If we put Isys /Iasym =5.0 for CH3OH into Eq.17, valid. But this may not be true for the less ‘rigid’ molec- we have r = 0.24 for 0 ≤ r ≤ 1; while if we put ular groups. Based on the successful treatment with the Raman Raman Isys /Iasym =0.54 for CH3CD2OH into Eq.17, there CH3 groups, we may assume that the same treatment on is no solution of r for 0 ≤ r ≤ 1 at all. This clearly shows the similarly ‘rigid’ NH3 and CH2 groups is also going that Eq.16 and Eq.17 do not generally give consistent r to work. If this approach fails to quantitatively interpret values, and certainly they can not be held true at the the SFG-VS spectra for a particular molecular group, one same time. may try to consider the possibility that this group might Now, with the corrected βccc/βaca value for CH3OH, have been very differently perturbed in the bulk phase the fact of missing CH3-as mode in SFG-VS from from at the interface. vapor/CH3OH interface can be readily addressed. From One clear advantage of this empirical approach is that 10

the problem of the Fermi resonance and combinational APPENDIX: Terms and expressions for C2v group mode couplings is automatically addressed through the corrections empirical correction. As we have known, to assess the effects of such accidental resonances and couplings can be a With the empirical approach derived and tested for the very difficult problem in ab initio calculations. Therefore, C3v groups, similar approach can be taken for the C2v the empirical treatment discussed here may provide a way groups. For them, bond additivity model has to be used 14 to quantitatively test the whole molecule approach. to obtain necessary βi′j′k′ tensor ratios for quantitative For example, one direct may is to try to use the whole calculation.18,19,25,26 molecule approach to calculate the polarizability tensors If the fixed coordinates system of the AB2 group with of the CH3 group in the CH3OH and CH3CD2OH, and C2v symmetry is defined with c axis as the symmetry compare the results directly with the IR, Raman and axis, ac plane as the A-B-A molecular plane, and b is SFG-VS data. Provided with successful applications to perpendicular to the molecular plane, then there are two a series of clearly studied molecular systems, one may A-B stretching vibrational mode with A1 mode as the expect that in the future ab initio computation can be symmetric stretching (ss) vibrational mode, and B1 mode generally applied for quantitative interpretation of non- the asymmetric (as) mode.10,54,55 The non-zero elements linear spectroscopy. for the stretching vibrational modes are as the followings. For the A1 (symmetric) mode:

SFG-VS βaac = Raβccc ; βbbc = Rbβccc (28) ′ ′ ′ ′ Raman: αaa = Raαcc ; αbb = Rbαcc (29) ′ V. CONCLUSION IR: µc (30)

where Ra and Rb are the two βi′j′k′ elements ratios. In summary, we discussed a total empirical approach For the B1 (asymmetric) mode: with symmetry analysis to quantitatively interpret observed SFG-VS spectra from molecular interfaces. In SFG-VS βaca = βcaa (31) ′ ′ such an approach, the Raman and IR tensor ratios are Raman: αac = αca (32) directly calculated from Raman and IR spectral measure- ′ IR: µa (33) ments using a certain correction procedure to the bond additivity model. This approach is tested with the SFG- Therefore, Eq.9 for IR intensity of the C2v group be- VS spectra from vapor/CH3OH and vapor/CH3CD2OH comes, interfaces.

There have been few attempts to use Raman and IR IR 2 I µ′ spectra in the condensed phase to quantitatively inter- sym c IR = ′ (34) pret SFG-VS spectra from molecular interfaces in detail. Iasym µa It has been demonstrated that the bond additivity model Eq.10-Eq.13 are all generally valid for Raman has been successful in quantitative interpretation of SFG- 10,54,55 VS of the CH and CH as well as H O groups for their ss measurement, and Eq.14 alone cannot be 3 2 2 used to determine both R and R values. Therefore, modes,10 but it has not been as successful with the as of a b these molecules or molecular groups. Through analysis bond additivity model has to be employed. of the problem we understood the empirical basis for the For the single A-B bond treated with C∞v symmetry, successes of the bond additivity model. So we proposed Eq.15 is generally assumed valid, and the definition of a0 to treat the failures of the bond additivity model with a and m0 of the single bond is the same as defines above. complete empirical approach. In this report we success- For the C2v group AB2 with the A-B-A angle as τ, we fully demonstrated with two examples that the empirical have the following: corrections to the bond additivity model can successfully interpret what the simple bond additivity model failed. 1+cos τ 1 This development can become a new addition to the ex- GA1 = + 10 MA MB isting methodologies in quantitative SFG-VS studies. 1 − cos τ 1 We believe that this empirical approach can provide new GB1 = + M M understanding of the effectiveness and limitations of the A B bond additivity model, and this report also provided a in which MA and MB are the atomic masses of the atom practical roadmap for its application in SFG-VS studies A and B; GA1 and GB1 are the inverse reduced masses of molecular interfaces. of the A1 and B1 normal modes, respectively, with the normal mode coordinates defined as,16,17,58 Acknowledgment. HFW thanks supports by the Nat- ural Science Foundation of China (NSFC No.20274055, 1/2 QA1 = (∆r1 + ∆r2)/(2GA1 ) No.20425309, No.20573117). ZFC thanks support by the − 1/2 Key Project of Ministry of Education of China (MOE QB1 = (∆r1 ∆r2)/(2GB1 ) No.204065). (35) 11

Then the bond additivity model gives the following with the r values calculated from the experimentally ob- ′ ′ expressions of the non-zero αi′j′ and µk′ tensors. tained Raman depolarization ratio ρ of the ss mode of For the A1 (symmetric) mode: the AB2 group using the following equation.

1 ′ 2 2 2 αaa = a0[sin (τ/2) + r cos (τ/2)](2GA1 ) 1 α′ = a r(2G ) 2 3 bb 0 A1 ρ = (37) 1 2 2 2 ′ 2 2 2 4+20(1+2r) /[(1 − r) (1+3cos τ)] αcc = a0[r sin (τ/2) + cos τ/2](2GA1 ) 1 ′ 2 µc = m0 cos(τ/2)(2GA1 ) Following the example of the C3v group above, the pro- For the B1 (asymmetric) mode: cedure to make correction for the bond additivity model is as the following. First to use Eq.37 to obtain r value. 1 1 ′ ′ 2 Then the Ra and Rb values can be obtained using this r αac = αca = a0[(1 − r) sin τ](2GB1 ) 2 value. In order to calculate Rc/a, one need to use Eq.8 1 ′ ′ ′ ′ ′ 2 with the empirically corrected α /α and µ /µ values. µ = m0 sin(τ/2)(2G 1 ) cc ac c a a B ′ ′ The µc/µa value can be obtained using Eq.34. Then the ′ ′ Therefore, the bond additivity gives the following βi′j′k′ αcc/αac value can be obtained by put the r value and the 10 ′ tensor ratios. bond additivity model expressions for αi′j′ tensors above into Eq.11 to derive a equation similar to Eq.18. This correction procedure is certainly worth to be βaac 1+ r − (1 − r)cos τ Ra = = tested for interpretation of SFG-VS data of typical C β 1+ r + (1 − r)cos τ 2v ccc molecules or molecular groups at interfaces. For example, βbbc 2r at the air/water and organic/water interfaces, so far the Rb = = βccc 1+ r + (1 − r)cos τ asymmetric stretching mode of the C2v water molecule 19,21,22 βccc βccc has not been clearly identiﬁed yet. Another exam- Rc/a = = βaca βcaa ple is that the intensity ratio between CH2-ss and CH2-as [(1 + r)+(1 − r)cos τ]cos(τ/2) G ω has been widely used to interpret CH2 group orienta- = A1 B1 (36) tional changes at molecular and polymer interfaces.65 (1 − r) sin τ sin(τ/2) GB1 ωA1

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