10064
J. Phys. Chem. 1993,97, 10064-10069
Orientation Analysis by Simulation of Vibrational Sum Frequency Generation Spectrum: CH Stretching Bands of the Methyl Group Cbiaki Hirose,' Hiroyoshi Yamamoto, Naotoshi Akamatsu, and Kamari Domen Research Laboratory of Resources Utilization, Tokyo Institute of Technology, 4259 Nagatsuta, Midori- ku, Yokohama, 227 Japan Received: November 30, 1992; In Final Form: May 25, 1993'
A description is given of the procedure which simulates the spectral features of vibrational sum frequency generation (VSFG)spectrum by the v(CH) stretching bands of the methyl group. Instead of assuming that cuff alone is significant as in previous expressions for the molecular hyperpolarizability tensor, we have introduced a parameter r = att/acfwhere all and at€are the parallel and perpendicular components, respectively, of the Raman tensor associated with C H bond. The statistical aspects of molecular orientation and finite width of an infrared light source were also incorporated. A model calculation is presented of the signals from twodimensionally arranged methyl groups to reveal the effects of the quantities on the spectral and angular profiles.
I. Introduction The deduction of the orientational order of surface species is an important applicationof vibrational sum frequencygeneration (VSFG) spectr~scopy,~-~ a method which exhibits unique advantages over methods such as IRAS (infrared reflectionabsorption spectroscopy)or HREELS (high-resolution electron energy loss spectroscopy)in its insensitivities to the material and physical environment of surfaces. The orientation of methyl groups located at liquid,4d glass,'-l0 metal oxide,l and ~emiconductor/metal5J2-1~ surfaces have been investigated through the analysis of the spectral profiles which were observed under specific combinations of polarization of the relevant lightwaves. Except for recent reports on the observation and analysis of the VSFG spectrum of Langmuir-Blodgett (LB) films of cadmium arachidate,9s10previous reports have dealt with methyl groups whose surface orientation is isotropic. The signals from such systems are independent of sample rotation, and the angle between the top axis and the surface normal (tilt angle) of the methyl group has been derived from the subsequent analysis. The formula used in such analyses contained two important assumptions: first, the hyperpolarizability tensor componentsof the methyl v(CH) modes were estimated on the presumption that a f tis the only nonvanishing component in the Raman tensor of the constituent CH bond, where {is the C-H bond axis; second, the statistical distribution of the tilt angle was neglected. In addition, previous reports did not mention the coefficients which relate the electric field associated with the SFG beam to a nonlinear source polarization but only theexpressions for nonlinear susceptibility tensors were the main subject in the analyses. The inadequacy of the first assumption became apparent in theoretical calculations of the Raman tensors of hydrocarbonsl5 and an appropriately revised formula was recently adopted to the analysis of the SFG spectrum observed for CH groups on the diamond (1 11)-( 1 X 1) surface.16 In this work, the components were taken as nonvanishing, where [ and are the a~ a,,,, coordinate axes perpendicular to the CH bond and the equality derives from the assumed axial symmetry of the surface CH bond. The Raman tensors were also calculated for methane, ethylene, ethane, cyclopropane, n-propane, t- and g-butane, t,tpentane, and cyclohexane's and inspection of the values reported indicates that tensor components other than aft must also be retained in deriving the Raman tensors associated with the CH To whom correspondence should be sent. *Abstract published in Advance ACS Abstracts, September 1, 1993.
0022-3654/93/2091- 10064$04.00/0
bands of the methyl and methylene groups, too. We thus need to have the revised expressions of the hyperpolarizability tensor components of the CH vibrations of these groups. As mentioned above, surface anisotropy was recently observed on the VSFG signalsof the methyl CH vibrations of multilayered cadmium arachidate LB films.9J0 Figure 1 shows the observed change of signal intensity as the three- (Figure la) and ninelayered (Figure 1b) sample plates were rotated about their surface normals. The orientationsrelating to the experimentalconditions at the film surface is shown in Figure 2. The intensitiesof the p-polarized component of the SFG signals, which were produced by the both gpolarized visible and infrared beams with the infrared light tuned to the degenerateCH vibration band, are plotted against the azimuth angle$ of the sampleplates. =' 0 corresponds to the situation when the plane made by the reflected and refracted SFG beams coincides with the direction of withdrawal of the plates from the liquid trough in their preparation. It is clear from Figure 1 that the methyl groups are oriented two-dimensionally on the surface. Furthermore, the angular width of the peak at $ = OD is apparently narrower in the nine-layered film than in the three-layered film, indicating that thestatistical degreeof theorientationdiffers with thenumber of layers. Similar features were also observed on the SFG signals of the symmetric stretch mode, too. In the present study, a revision is made of the formulas which simulate the spectral and angular profiles of the VSFG signal. In section I1 we give the formulas which are the basis of the present study. The revision of the hyperpolarizability tensor of methyl group is described in section 111. The formulas obtained for methylene (CH2) group aregiven in the Appendix. In section IV, an account is made of the finite width of the infrared light and the statistical aspects of molecular orientation. Section V presents and analyzes the results of model calculations. We use four-letter abbreviations to designate specific SFG signals; the first letter, s or d, specifies whether the vibrational band under consideration is associated either with a symmetric (s) or degenerate (d) mode, and the ensuing three capital letters denote from left to right the polarizationsof the SFG, visible, and infrared beams. Thus the term sSPS signal stands for the s-polarized component of the SFG signal which is produced by the p- and s-polarized visible and infrared beams, respectively, with the infrared beam tuned to the peak of the symmetric vibration band. Using this notation, Figure 1 shows the plots of dPPP signals.
+
0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 10065
Orientation Analysis by SFG
corresponds to its absence, x1jkm: 000 0 0
(3) 0
0
In the absence of an electronic resonance of the surface species, X I , ~ , ~ values 'S are characteristic to the surface itself while xIjk,r)s values reflect the vibrational and orientational characteristics of the surface species. Equation 4 shows that xjjk,r is the sum of the molecular hyperpolarizability tensor components, &jk,r, where the sum is taken over the molecules participating to the SFG process, and the sum may be replaced by the ensemble average multiplied by the total number, N, of the molecules:
0
0
(b) 00
0
0
0 0
where ( ) denotes the ensemble average. We can express j31jk,r as a linear combination of the molecule-fixed paB7's:
0
0
)o o o o o p
(4) 0 0
qoo, goc
I
1
0 +I( 9 /degree Angular profiles of the SFG signals observed from the three-
Figure 1. (a) and nine-layered(b) LB filmsof cadmiumarachidate. Thefrequency of the IR beam was tuned to the degenerate CH stretch band, and the incident angles of both ppolarized visible and IR sources were 40° and 4 5 O , respcctively. The azimuth angle is the angle between the surfaceprojected propagation direction of the SFG beam and the direction of the withdrawal of sample plates from liquid trough in their preparation.
+
(5)
where a,0, and y are the coordinate axes of the molecule-fixed (abc)system. Explicit expressions for the coefficientsufjk.47are presented in Table 11 of ref 3. is a sum of terms associated with the vibrational modes of the surface species:
where
In eqs 6a and 6b, u denotes thevibrational mode of surface species, rUare the characteristicfrequencyand homogeneouswidth, respectively, of the mode, and OIR is the frequency of the infrared light source. We can express & - 4 7 ~ as 1 ~the product of the Raman tensor component and vibrational transition dipole moment: o,and
Figure2. Layout of SFG,visible, and infrared beams at the film surface. The azimuth angle (see the caption of Figure 1 for the definition) is also indicated.
+
11. Summary of Basic Formulas3
(7) When we assume the methyl group to have C,, symmetry, the following components are nonvanishimg:
Denoting by I, J, and K the coordinate axes of the laboratoryfixed (XYZ) system, the p-polarized component of the sumfrequency field EpSFis related by eq l a to the [component of the
source polarization PI(^), which in turn is related by eq l b to the J and K components of the visible and infrared light fields, and EdR, respectively. The coefficient4)is an optical parameter determined by the incident angles and the refractive indices. The ratio &,$/bywas 1.8 for the experimental condition that gave the results shown in Figure 1. X I J ~ laboratory-projected ~ , component of the macroscopic nonlinear susceptibility tensor associatedwiththeSFGprocess, isrelated tothesurface-projected components xIlkSF as follows:
where i, j , and k are the coordinate axes of the surface-fixed (xyz) system, and the transformation coefficients U I J ~ ;are U~S functionsof the dihedral angle of the X I plane with respect to the plane of incidence, the XZ plane. xukSF may be written as a sum of two terms, one that corresponds to the presence of vibrational resonance, ~ ~ j kand , ~ the , other that
where the subscripts s and d denote symmetric and degenerate vibrational modes, respectively. It is seen from eq 6 that the same relations hold for &@7,v(sand &-,,471S and also the substitution of the equivalents of eq 8c into eq 5 and the use of Table I1 of ref 3 reveal that the contribution by the Dam component and its axial counterparts disappears when the torsional angle of the methyl group is randomly distributed or when the methyl groups are under free internal rotation. 111. Revision of B47,rR
A consideration of the physicochemicalenvironmentof methyl and methylene groups suggests that the Raman tensor associated with the CH bonds have C, symmetry. We have then five components, namely, ate, am,arc, and a~ = uef (the 4 and {axes are taken to lie on the symmetry plane) instead of the three components, arc and = a,,,,, of axial symmetry. Equation 24 of ref 17 is then replaced by properly revised expressions. It is almost impossible, however, to estimate the values of the four independent components of the CH Raman tensor from the available data, and we proceed by assuming the CH bonds to be
Hirose et al.
10066 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
axially symmetric in the belief that the relations la(€- awl,lad < Jaw awl < Iafi hold and rationalize this assumption. We thus have the simplified relations that
+
ace= a*), a{(= 0
(9)
and introduce the parameter r:
r = ‘y&ff
(10)
TABLE I: Values of Parameters Used To Calculate Theoretical Curves Molecular Constants and Parameters vibration frequency (hwhm) Y,,,,, = 2878 cm-l (5 cm-l) Y@ = 2960 cm-* (5 cm-I) U F= ~ 2945 ~ cm-I (5 cm-l) b2 = 0.35 (mixing coeff of Fermi interaction) r = 0.14
The relative intensity of the Raman bands, IspymRlmaa/zdcgRaman, and the depolarization ratio of the symmetric mode, ply,,,, are then expressed as follows:18
Optical Parameters visible angle of incidence, deg index of refraction fwhm of IR source: 13 cm-1
p,,
= 0.75/ [ 1 + 11.25( 1 + 2r)*/( 1 - r)2]
(1 1b)
when a linearly polarized excitation source is used for the Raman spectroscopy. An inspectionof the availabledata19indicates that r lies within the range from about 0 to about 0.25 for simple halogenoalkyls and carboxyl acids. The procedure described in ref 17 gives eqs 12 and 13 as the revised expressions for @a~7,VRto replace eqs 25 and 26 of ref 17.
40 1.460
infrared
SFG
45 1.412
1.465
Orientation-Related Parameters = *45O (angle of surface-projectedtop axis from surface-fixedx axis) B = -35O 20° (tilt angle of top axis from surface normal) 7
*
We assume the frequency profile of the infrared source to be Gaussian. For an infrared source operating with frequency centered on WIRO, the field amplitude E(WIR) at frequency OIR is given by
For the symmetric vibration band: = (Au[R)fwhm/(2 In 2,
where 1 3 1 is~ the width parameter, and (AqR)fwhm is the full width at half-maximum. The frequencyspread of thevisible light source is generally much less than the vibrational width rvand may be neglected. The SFG field at the nominal frequency of WFO = wVis WIR’ is now spread around WSF’ by a width of ( h I R ) h h m , and the signals detected are the convoluted sums. Figure 3 depicts how the calculated spectral profiles change by making allowancefor the finite frequencywidth of the infrared source. The vibrational bands at 2878 cm-l (symmetric CH stretch, band), 2960 cm-I (degenerate CH stretch, U d s j band), and 2942 cm-l (first overtoneof the HCH deformation, Y F band) ~ ~ were assumed to have a common bandwidth (hwhm) of 5 cm-1, as estimated from the FT-IR spectrum of the nine-layered sample, and a value of .0.35 was assumed as the squared Fermi resonance c o e f f i ~ i e n t b When ~ . ~ ~ thefinitewidthoftheIRsourceisignored, the calculation gave the profile shown in Figure 3a but the introduction of the frequency spread (13 cm-l in fwhm) of our IR source resulted in the profiles shown in Figure 3b,c. Figure 3b shows the squares of the convoluted sum of the SFG field amplitudes, the profile expected when the IR source is coherent over the frequency spread. Figure 3c is the profile expected from an incoherentlybroadened ir source. The actual profile observed from monolayered LB films of cadmium arachidate is shown in Figure 3d. The observed profile has a closer resemblance to the profile shown in Figure 3c, and the subsequent calculations were made by convoluting the squared amplitudes. The signal profiles are also affectedby the statistical distribution of the spatial orientation of the surface species. The orientation of a molecule is defined by a tilt angle 8, which is the angle made by the molecule-fixed c and the outward normal of the surface, theinclination (or azimuth) angle T,theangle between thesurfaceprojection of the c axis and the surface-fixed x axis, and the torsional angle 4 which is the dihedral angle of the moleculefixed ac plane with respect to the surface. For vibrational SFG to be observed, a preferred tilt angle is essential. Variation of signal intensity with spatial rotation of sample plate occurs when a preferred direction of molecular inclination is added to the tilt. A fixed torsional angle leads not only to a change in the angular profiles but also to a splitting of the degenerate C H stretch band. We assume that the distribution of an orientational angle a (astands for 8, x . or 4) obeys the Gauss distribution law. The
+
where T is the HCH bond angle, expressions for G,, are given in ref 17, and
and Gdeg
In eq 13, a0 is a constant coefficient and ulym and Udcg are the vibrational quantum numbers. A model calculation was made for the range from r = -0.2 to +0.2 by using the values listed in Table I for other parameters and by assumingthat alkylchains are in an all-trans conformation with an orthorhombic surface orientation. The results showed that increaseof r leads to an increaseof the relative signal intensity of the symmetric band with respect to the degenerate band but no significant change in the angular profiles.
IV. Statistical Aspects Thespcctral featuresofSFG signalareaffected by thestatistical characteristics contained in molecular, optical, and orientationdefining quantities. The statistics involved in molecularquantities are the widths of the vibrational bands. We assume them to be homogeneous and described by l?” as shown in eq 6. rumay be estimated from the widths of infrared absorption bands. The possible presence of inhomogeneous broadening may be covered by the method of Bain et al.14
(14)
Orientation Analysis by SFG
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 10067
(i, / degree
I
Figure 4. Angular profiles of SFG signals for different tilt angle 8. The results for 0 = -30' (a, b) and -50 (c, d) are shown. A negative value of 8 impliesthat the top axis is inclinedaway from the direction of surfacefixed x axis. An all-trans conformation of the alkyl groups with orthorhombic orientation on the surface was assumed. The profiles of the dPPP (solid), sPPP (broken), and dSPP (dotted) signals are shown in (a) and (c), and those of sSSP (solid), dSSP (broken), and dPSP (broken) signals in (b) and (d). The curves are normalized to the dPPP ($ = Oo) signal in (a) and (c), and to the sSSP($ = 180') signal in (b) and (d).
0
0 Figure 3. Calculated (a+) and observed (d) SFG intensity profiles of methyl u(CH) bands. The intensityis plotted against the center frequency OIR' of the infrared beam for the cases where the frequency spread of the ir beam is negligible (a) and 13cm-' in fwhm (b and c). All of the light beams were p polarized. Figure 3b is the plot of the squares of the convolutedsum of the SFG field amplitudes (the IR light is coherent over the spread) and Figure 3c is the plot of the convoluted sum of the squared field amplitudes (the IR beam is incoherently spread). The assumed frequencies of the vibrational bands were uSum= 2878 cm-l, u h = 2960 cm-I, and up-i = 2942cm-I . Figure 3d showsthe SFG spectrum observed from the monolayered LB film of cadmium arachidate.
number of molecules N ( a ) which are oriented by a is then given bY
N(a)= Nda)
(15)
6, = (aA)/ln 2
(16)
where
a0 and ha are the center and the width (hwhm), respectively, of the distribution and NOis the total number of the relevant molecules. The transformation coefficients uilk,,t7 in eq 5 are associated with the orientational angles and given in the form of trigonometricfunctionssin (na)andcos (na) ( n = 0-3, seeTables I and I1 of ref 3). The averages of these functions are derived by using Laplace integrals as follows:
(sin(na)) = J:sin(na)fla)
da =
cos(nao) exp(-n262/4) (17a) (cos(na)) = J:cos(na)fla)
da = sin(na,) exp(-n262/4) (17b)
It is noted here that the Gauss distribution function f(x) is defined in the range x = ---+a while the present variables are periodic,
and eqs 17a and 17bare physically meaningful, to 99% reliability, when Au < 85O (for the case that -180O < a < +180°). We portray the orientational features of molecules, Nt in total number, as below. (1) Denoting by Rdmthe fraction of the molecules which are random in three dimensions, the SFG signal arises from the remaining NO= Nt(1 - R h ) molecules. The NOmolecules are assumed to be tilted by the angle distributed over 00 f A& (2) Wenext introducetheparametersRha"andRWrtodescribe the randomization and reversal, respectively,of the azimuth angle T . N,$dmr molecules are two dimensionally random, and of the remaining NO(1 - Rdmr)molecules, NO(1 1 - RWr) are inclined in the range specified by TOf AT while half of the residual No( 1 - Rdmr)Rrwrmolecules are inclined by ( T O f AT) and half by ( T O f AT) 180O. R-T loses its meaning when R ~ = T 1. (3) Rrdm*and Rm* are defined to describe the randomization and reversal, respectively, of the torsional angle 4. Here, the randomization originates either from the free internal rotation of the methyl group or from the random distribution of 4. NO(1 - Rd,*)(l-R,*) molecules out of the torsionally frozen NO(1 - Rrdm*)molecules have their torsional angles distributed within +& f A+ while of the remaining NO(1 - Rh*)R,* molecules half have their ac planes inclined to the surface by +r#~,, f A4 and half by+o f A$. The reversal of 4 is envisioned as the switchover of an HCC plane (ac plane) between the + y and -y directions. Rrcv*loses its meaning when Rdm* = 1, the case encountered when the experiment is made at room temperature, or when 4 = Oo, 120°, or 180O. Angular factors are incorporated into the simulation program through eqs 5 and 17, and the randomization and reversal factors may be accounted for in the calculation by eq 4 of the macroscopic susceptibilitytensor via the average or sum over the contributing molecules.
+
V. Discussion The effects of orientational parameters and their distribution widths were examined by model calculations in which the value of one of the parameters was varied while other parameters were kept at the values cited in Table I. A rhombic orientation of the surface-projected top axes was assumed to accommodate the simulation of the observed profiles shown in Figure 1, and the well-oriented methyl groups were given the azimuth angles of +r and -T, in pairs. Figures 4-7 show the changes in the angular profiles calculated for various values of tilt angle 0 (Figure 4),
Hirose et al.
10068 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
- 180
0
-180 0
0
+180 -180 (i, / degree
0
(i,*180 / degree -180
180
*180
Figure5. Angular profilesof SFG signals for variousvaluesof distribution width of 0. The calculated profiles for 0 = -35' f 10' (a, b) and 0 = -35' 30' (c, d) of thedPPP (solid), sPPP (broken),and dSPP (dotted) signals are shown in (a) and (c), and of the sSSP (solid),dSSP (broken), and dFSP (dotted) signals in (b) and (d). See also the caption of Figure
*
4.
...... 3 0.0
4/
degree
Figure 7. Angular profiles of SFG signals for various degrees of twodimensional randomization, RdmT.The calculated profiles for R h T= 0.0 (a, b), Rdmr= 0.2 (c, d), R h r = 0.4 (e, f), and R h T = 0.6 (g, h)
of the dPPP (solid),sPPP (broken),and dSPP (dotted) signals are shown
in (a), (c), (e), and (g), and of thesSSP (solid), dSSP (broken),and dPSP
(dotted) signals in (b), (d), ( f ) , and (h). An orthorhombic orientation ( T = 45') with 0 = -35' was assumed for the methyl group. See also the caption of Figure 4. 10 0
-160
0
.Id0
dJ /
-180
.
.1
eo
degree
Figure 6. Angular profiles of SFG signals for different values of the azimuth angle T . The alkyl chains were assumed to be in a rhombic orientation with +T and -T orientations, in pairs. The profiles for T = 30' (a, b), T = 45' (c, d), and T = 60' (e, f ) of the dPPP (solid), sPPP
(broken), and dSPP(dotted) signals are shown in (a), (c), and (e), and of the sSSP (solid),dSSP (broken),and dPSP (dotted)signals in (b), (d), and ( f ) . See also the caption of Figure 4.
its distribution width A0 (Figure 5 ) , and azimuth angle T (Figure 6), and the degree of two-dimensional orientation RdmT(Figure 7). The curves are normalized to the maximum intensities of the dPPP and the sSSP signals. Figure 4 reveals that both the relative intensity of the sPPP signal with respect to the dPPP signal and the angular profiles ofall thesignalschangewithe. Figure 5 indicates that the primary effect of the distribution width (A6') is to change the angular profiles of the dSPP and dPSP signals. It is seen from Figure 6 that the azimuth angle T affects the angular profiles of the dSPP, dPSP, and dSSP signals, but the profile of the dPPP signal is little affected. The effect from the distribution width AT (not shown) was found to be negligible up to f20°. Figure 7 indicates that both the relative intensity and the angular profiles of all the signal components are functions of RrdmT. To summarize the features of practical value arising from the simulation, the relative intensity of the sPPP signal at J. = 180° withrespect tothedPPPsignalintensityatIC, = Oo isquitesensitive to the orientational parameters, to Rrdmr in particular. Also sensitive are the ratio of the dPPP (J. = 1 8 0 O ) to the dPPP (J. =0 ' ) signal intensities and the angular spread of the dPPP signal around the maximal point at J. = Oo. The parameter r, which
5.0
00
0.0
00
010 r
020
0
20
40
@/degree
60 A@ /
degree
Figure 8. Dependenceon parametersr(o), 0(b),and Ae(c) of the relative SFG intensity of randomly oriented methyl groups ( R h T= 1.0). The
intensitiesof the sPPP (O), sSSP ( O ) ,and dSSP (0)signals (normalized to the dPPP signal intensity) are plotted against r (at 0 = -35'),0 (with A0 = ' 0 and r = 0.14), and A0 (at 0 = -35' and r = 0.14), in (a), (b), and (c), respectively. is the ratio ( ~ ~ ~ / ' ~ther rCH o f Raman tensor, was found to affect the relative intensity of the sSSP and sPPP signals with respect to that of the dPPP signal. The tilt angle and the azimuth angle were found to affect the overall features in a rather complicated manner. Put differently, our calculations reveal the need for the angular profiles of more than two different SFG signals, e.g., the dPPP, sSSP, and dSSP signals, with the angular interval of about 5 O , in order to determine the orientational order of the methyl groups. The relative SFG intensity of the vlYmband with respect to that of the vdeg band is widely used to derive the tilt angle of twodimensionally isotropic methyl groups. Figure 8 shows the r, &, and A8 dependences of the relative intensity as calculated for the case when Rdmr= 1.O. The intensities of the sPPP ( 0 )and sSSP ( 0 )signals as well as the dSSP (0)signal, all normalized by the intensity of the dPPP signal, are plotted for various values of r (Figure 8a), 80 (Figure 8b), and A0 (Figure 8c). It is seen that
Orientation Analysis by SFG
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 10069
the effects of r and A0 are smaller than that of 00 itself but cannot be disregarded if accurate estimations of 00 are to be made. In conclusion, the electric field vector of sum frequency beam and the source polarization vector thereof have been related to the molecular susceptibilitytensor attached to methyl group (eqs 1-8). The variation of SFG signal by the azimuthal angle of sample plate originates from the two-dimensional anisotropy of macroscopic SFG tensor and the anisotropy in turn arises from the ordered orientation of molecules on the surface. Two sets of Euler angles, one defining the orientation of sample plate with respect to the laboratory coordinate system and the other specifyingthe orientation of molecules with respect to the sample surface, are required to analyze the observation. The components of molecular susceptibility tensor which are related with the components of linear vibrational dipole moment vector and Raman scattering tensor (eqs 6 and 7) have been further related to the quantities attached to the CH bond of methyl group so that the relative magnitudes of the components are estimated (eqs 9-1 3). Procedures for the incorporation of the orientational, molecular, and optical parameters acompanied by finite distribution widths into the formula for the VSFG signal have been described. Model calculations disclose the intricate effects of these on the spectral and angular profiles of the signal intensity. The application of the present procedure to the analysis of the experimental results will be described in a forthcoming paper.1° The procedure described in the present paper is expected to help in choosing the sets and intervals of experimental variables for the significant estimation of the orientational parameters. Omitted from the present treatment are the possible Occurence of discrete but different values of 00 and TO and allowance for different adsorption sites, microdomain formation, and the multiplicityof adsorbate conformations,all of which may be extant in real systems. They may be handled in principle through appropriate modification of eqs 2 and 4-7.
Acknowledgment. The authors are deeply indebted to Prof. S. F. A. Kettle of University of East Anglia for his critical reading of the manuscript. The present work has been supported by the Grant-in-Aid for Scientific Research (A) awarded by The Ministry of Education, Science and Culture of Japan, No. 04403004. Appendix. Revised Expressions for the Hyperpolarizability Tensor Components of the CH Vibrations of the C H z Group The revised expressions for the hyperpolarizability tensor componentsof the methylene group have been derived. Assuming the C-H bond to be axially symmetric, the relative intensity of the Raman signals,&lRomn/~blRo", and the depolarization ratio pal of the symmetric CH mode are expressed as follows:18 zalRPman/~blRaman= 0.475
pol = 0.75/[1
+ 2.036( 1 + 2r)'/(1
- r)'
+ 7.5(1 + 2r)2/(1 - r ) 2 ]
(Al) (A2)
where the subscripts a1 and bl denote the symmetric and antisymmetric CH stretch vibrations, respectively.
The tensor components fld7,,RSare found to be as follows: For the a, band:
where a is the HCH bond angle. The c axis is taken along the bisector of the HCH angle and the ac plane as coinciding with the HCH plane; tuand t b are quantities defined similarly to eq 14a:
References and Notes (1) Shen, Y. R. The Principles of Nonlinear Optics; Wiley: New York, 1984; Chapters 6 and 25. (2) Hcinz, T. F. Nonlinear Surface ElectromagneticPhenomena;Ponath, H.-E., Stegeman, G. I., Eds.; Elsevier: New York, 1991; Chap. 5, pp 353415. (3) Hirose, C.; Akamatsu, N.; Domen, K. Appl. Spcrrosc. 1992, 46, 1051. (4) Guyot-Sionncat, P.; Hunt, J. H.; Shcn, Y. R. Phys. Rev. Lett. 1987, 59, 1597. (5) Miragliotta, J.; Polizzotti, R. S.;Rabinowitz, P.; Cameron, S. D.; Hall, R. E. Chem. Phys. 1990, 143, 123. (6) Superfine,R.; Huang, J. Y.; Shcn,Y. R. Phys. Rev. Lett. 1991,66, 1066. (7) Guyot-Sionnest, P.; Superfine, R.; Shen, Y. R. Chem. Phys. Lett. 1990,144, 1. (8) Huang, J. Y.;Superfine,R.;Shen,Y.R. Phys. Rea 1990, 42A, 3660. (9) Akamatsu, N.; Domcn, K.; Hirose, C.; Onishi, T.; Shimizu, H.; Masutani, K. Chem. Phys. Lett. 1991, 181, 175. (10) Yamamoto,H.;Akamatsu,N.;Domen,K.;Hirose,C. J.Phys.Chem., following paper in this issue. (1 1) Hatch, S. R.; Polizzotti, R. S.;Dougal, S.;Rabinowitz, P. Chem. Phys. Lett. 1992, 196, 97. (12) Hams, A. L.; Chidsey, C. E. D.; Levinos, N. J.; Loiacono, D. N. Chem. Phys. Lett. 1987,141, 350. (13) Harris, A. L.; Rothbcrg, L.; Dhar, L.; Levinos, N. J.; Dubois, L. H. J. Chem. Phys. 1991,94, 2438. (14) Bain, C. D.; Davies, P. E.; HuiOng, T.; Ward, R. N.; Brown, M. A. bngmuir 1991, 1563. (15) Gounh. K. M. J. Chem. Phvs. 1989. 91.2424. (16) Chin;R.P.;Huang,J.Y.;Shen,Y.R:;Chuang,T.J.;Seki,H.;Buck, M.;Phys. Rev. B 1992, 45, 1522. (17) Hirose, C.; Akamatsu, N.; Domen, K. J. Chcm. Phys. 1992,96,997. (18) Long,D. A. RamanSpectroscopy; .. McGraw-Hill International: New York, 1977. (19) Schrader, E., Merier, W., Eds.DMS RamanllR Atlas of Organic Compounds; Vcrlag: Weinheim, 1974.
