Raman scattering cross sections and polarizability derivatives of H2S, D2S, and HDS

JOURNAL

OF MOLECULAR

SPECTROSCOPY

156,431-443 ( 1992)

Raman Scattering Cross Sections and Polarizability Derivatives of H2S, D2S, and HDS’ J. M. FERNANDEZ-SANCHEZAND W. F. MURPHY Steacie Institute,for Molecular Sciences, National Research Council qf Canada, Ottawa. Ontario, Canada KIA OR6 Absolute Raman scattering cross sections have been measured for the trace and anisotropy scattering of gaseous H2S and several mixtures containing varying amounts of H2S, D$. and HDS. The trace scattering cross sections have been interpreted in terms of a set of two isotopically invariant &G/G, parameters and the isotopic mole fractions. For the anisotropy cross sections, the analysis is complicated by the fact that there are two polarizability components which contribute to each of the totally symmetric modes, together with the fact that the bands due to the two stretching modes overlap in the spectra of H2S and D$. Relations between these polarizability components were determined from analyses of ro-vibrational band profiles in the H,S spectra. These relations were used as constraints in the analysis of the anisotropy cross sections, to obtain approximate values for each of the polarizability component derivatives. rzl1992 Academic PRESS.IIK.

1. INTRODUCTION

In the past few years, there have been several reports of intensity studies of the Raman spectrum of hydrogen sulfide. Gaufres and co-workers have measured the depolarization ratio of the Rayleigh band (1) and have considered the profile of the pure rotational Raman scattering (2-4). In the latter work, they found that values of the ratio of the equilibrium polarizability anisotropy components which differ by up to a factor of three were required to reproduce different regions of the spectrum. Other studies of the pure rotational spectrum of hydrogen sulfide are reported in Refs. ( 5 ) and (6). The absolute intensity of the ul band ( 7,8) and the rotational profiles of the bending band and the SH stretching region have also been considered ( 9-1 I ). The rotational profiles of the totally symmetric vibrations, like that of the pure rotation, are complicated by the fact that the transition intensities include contributions from two polarizability tensor components (5, 12, 13); their relative value is one of the parameters which determines the profile. Furthermore, the stretching region is affected by Coriolis coupling between levels of the symmetric and asymmetric stretching modes. Each of these factors must be taken into account in the analysis of the observed spectra. Over the years, the vibration-rotation hydrogen sulfide spectrum has posed an intriguing problem for molecular spectroscopists. Early efforts at fitting observed transition frequencies in terms of model Hamiltonians were only partially successful. A satisfactory fit was obtained only when it was realized (14) that the numerical system is better behaved when an I’ (prolate top) molecular axis system is used, in spite of the fact that the molecule is a near-oblate asymmetric top. On this basis, the highresolution absorption spectra of the pure rotation ( 13 ) , stretching ( 16), and bending ’ Issued as NRCC

333 11

431

0022-2852/92 $5.00 Copyright(i 1992 by Academic

Press. Inc

All rights of reproduction in any form reserved

432

FERNANDEZ-SANCHEZ

AND MURPHY

( 17) regions of H2S have been analyzed. Spectra of the D2S isotopic species have been similarly considered (18, 19). Other evidence for the unconventional nature of this molecule was found in the intensity analysis of the absorption spectra of the stretching region (16). To satisfactorily describe the detailed intensity behavior, an 18term transition moment expression was used, in which several of the higher order terms were in fact larger than one of the constant, first-order, terms. The possibility of similar behavior must be kept in mind during the analysis of the rotational structure in the vibrational Raman spectrum. When considering the profile of the H2S pure rotational spectrum, it must be noted that the anisotropy of the equilibrium polarizability is very small (3. ZO), so that normally negligible vibration-rotation interaction effects may be large enough to be observed in this case; indeed, such explanations have been suggested previously (4, 21). In any consideration of the effects of vibration-rotation interaction on the rotational intensities, it is necessary to have reliable, self-consistent values for the intensity parameters for the vibrational bands. Previous determinations of such quantities have been hindered by the use of spectra measured under relatively low-resolution conditions, and, in some cases, by the use of unnecessary simplifying assumptions. There are available, however, ab initio calculations of the polarizability components and their derivatives (22), which are useful as a guide to expected magnitudes and, more importantly, absolute signs that are not obtainable from the analysis of the intensities. Thus, we have reconsidered the Raman intensities of the hydrogen sulfide vibrational bands and have measured absolute trace and anisotropy cross sections for the H2S, D$, and HDS isotopic species. We have also considered the dependence of the rotational profiles of the H2S bending and stretching regions on the relative values of the polarizability tensor components. In this way, we have produced a consistent set of intensity parameters for the vibrational Raman spectrum of hydrogen sulfide, which can be used with some confidence in the consideration of the pure rotational spectrum. The latter work is presented in the accompanying article (23). II. METHODS

AND

MATERIALS

The sample of H$ (electronic grade), with a specified purity of 99.99%, was obtained from Scientific Gas Products, Inc. The sample of D$ (97 atom% D) was obtained some time ago from MSD Isotopes and contained some DZ (presumably due to reaction of D$ with the iron cylinder), which was removed by trap-to-trap distillation with liquid NZ. No other impurity, apart from isotopic impurities, was observed in the measured spectra. Several mixtures of H2S/D2S were prepared (at approximately 20, 40, 60, and 80 atom% D) in order to “synthesize” the intermediate species HDS, which cannot be isolated. The mixtures were liquified to ensure that H/D exchange took place to the highest extent. After that, no effects due to changes in the mixture composition could be detected in the recorded spectra. A cylindrical fused-silica sample cell with Brewster-angle windows was loaded with a mixture of hydrogen sulfide and nitrogen at partial pressures of about 300 and 450 Ton-, respectively. The Raman scattering intensity was enhanced by the use of our multipass system (24). The room-temperature gas-phase Raman spectra were measured with a Spex 140 18 double monochromator with a cooled Burle C3 1034 photomultiplier tube and a Spex Datamate digital data acquisition system. The spectral slit width was about 2 cm-’ and 3 or 4 scans were accumulated at a sampling time of 1 set/point for the stretching regions and 3 set/point for the bending regions. The samples were

RAMAN

INTENSITY

OF HYDROGEN

4.33

SULFIDE

excited with about 6.5 W of Art laser radiation at 5 14.5 nm, from a Coherent Innova 200. Parallel and perpendicularly polarized components of the Raman spectra were recorded. Wavenumber shift and spectral sensitivity corrections were applied, and the trace scattering spectrum was calculated from the polarized components in the usual way (25). Wavenumber shift values are estimated to be accurate to + 1 cm--l . As in our previous work (26, 27). the molar cross section of each band relative to that of the Q branch of nitrogen was calculated by multiplying the ratio of the band areas by the ratio of the nitrogen/ hydrogen sulfide partial pressures. The bending modes fall in a region where there is a nonvanishing polarized background from the cell walls that was subtracted before integrating the band areas. The trace scattering cross sections were then calculated from the corresponding value (40.8 + 0.8 ) X 10 mm36 m’/sr for the N2 Q branch with 5 14.5-nm exciting radiation (26). Since in the trace spectra the bands are well isolated, the error for the reported cross sections is estimated to be 5%, with a minimum error of kO.2 X 1Om36m’/sr, as was done previously in the case of acetylene (28). The anisotropy cross sections were obtained by measuring the band areas of the perpendicularly polarized component, scaling them by 7/3, and then proceeding as for the trace cross sections. Since the anisotropy spectra had to be corrected for the contribution of the N2 spectrum in the SH stretching region, and for that from the silica cell walls in the bending region, we have chosen for the anisotropy cross sections the more conventional error estimate of 10% of the reported cross section, with a minimum of +0.5 X 1O-36 m’/sr. The observed Raman trace and anisotropy scattering differential cross sections obtained in this way are reported in Tables I-III. III. CALCULATION

METHOD

A. Trace Spectra

The approach used in the intensity analysis was presented previously (26) and is only outlined here. In the double-harmonic approximation. the absolute differential TABLE Calculated

I

and Observed Gas-Phase Raman Trace Scattering Differential Cross Sections Excitation) for Isotopic Species of Hydrogen Sulfide”

( 514%nm

Assignment Hand

position. CIll-’

HIS

D>h

otwrvcd

IIDS

(‘al< Ilked

1’2

O.Oi

0.11f0.2

I .2!1

1 .XiO.’

1902. 1’1

1x95. 1034. 856. mole

v2

fraction

a Cross

sections

fraction

is the estimated

has

heen

0.0563

0.9437f0.0030

in units

constrained

of 10-36m2/sr. standard

in the

For

deviation.

lit to give a total

the

DzS

propagated mole

results,

the

error

in the fit, whilr

fraction

of unity.

limit the molr

for the fraction

DzS

mole

of HDS

434

FERNANDEZ-SANCHEZ

AND

MURPHY

TABLE II Calculated and Observed Gas-Phase Raman Trace Scattering Differential Cross Sections (5 145nm Excitation) for Some Isotopic Mixtures of Hydrogen Sulfide, with the Mole Fractions Derived in the Fit” Band position c1n-’ 2620. 2613.

Assignment H,S

DzS

v3 -

11,

1902.

856. Isotopic

1’2

232.3

-40%

ObS

D

Calc

-60% Obs

D

Calc

-80% Obs

D

Calc

Obs

235.2 jz11.8

171.8

171.2 zt8.6

122.4

122.2 zt6.1

66.2

66.3 f3.3

53.9 f2.7

105.1

104.8 f5.2

146.8

146.0 k7.3

194.3

195.2 f9.8

53.9 0.73

0.66zkO.2

0.41

0.42j~O.2

0.23

0.21z!zo.2

0.05

“2

0.50

0.52f0.2

0.66

0.68*0.2

0.62

0.62f0.2

0.50

0.56&0.2

-

0.03

0.12zto.2

0.23

0.26f0.2

0.48

0.50*0.2

0.80

0.88*0.2

u2

1034

D

Calc

1’1

1’1

1895. 1183.

-20%

HDS

0.10*0.2

mole frart,ions H?S

0.59810.062

0.336zkO.064

0.186f0.064

DzS

0.02OztO.064

0.166+0.066

0.349kO.066

0.585f0.064

0.381

0.498

0.465

0.375

HDS

’ Cross sections in units of 10F3” m*/sr. Thr error limits for the mole fractions are r&mated standard in the fit. III racb case, the HDS mole fraction is constrained to give a total mole fraction of unity.

0.040*0.062

deviations,

propagated

Raman trace scattering cross section for the ith normal mode is a function of the derivative of the mean polarizability for that mode, and is given in the SI system by (29)

where to is the permittivity of vacuum (=8.8542 X lo-” C V -’ m-l), qi is the dimensionless normal coordinate associated with the harmonic frequency w, , C5is the wavenumber of the scattered radiation, and h, c, k, and T have their usual meaning. 66 /dqi is one-third of the trace of the derivative of the molecular polarizability tensor with respect to qi , i.e., the mean of the diagonal Cartesian tensor components. In the SI system, d& /dq, has units of C m2 V -’ (29). This derivative can be expanded in terms of the symmetry coordinates S,, z=

I

$L,.,

(2)

J

where L,, are the elements of the normal coordinate matrix obtained as the eigenvectors in the force field calculation. In a nonlinear optimization procedure, initial estimates of the isotopically invariant parameters da /LJS, are adjusted to fit the observed trace scattering cross sections for hydrogen sulfide. The only nonvanishing parameters are those for the symmetric SH stretching ( S1 ) and for the SH2 bending ( S2), the symmetry coordinates which transform according to the totally symmetric representation of the C2c point group of the parent H2S molecule. Each cross section is given a weight proportional to the inverse of its estimated error. The mole fractions of the various components were included as fitting parameters, so that the abundances of the isotopic species were also determined in the fit of the observed cross sections. We use an error propagation technique (30) to estimate the parameter error limits.

RAMAN INTENSITY OF HYDROGEN

435

SULFIDE

TABLE III

Calculated and Observed Gas-Phase Raman Anisotropy Scattering Differential Cross Sections (5 145nm Excitation) for Some Isotopic Mixtures of Hydrogen Sulfide” i\ssignment Rand

position,b

cm-’

HDS

Calculated’

Observed

H?S sample 178.0

172.4

26.58.

26.52

1220.

f17.8

Zfi.56i2.66

D2S sample 1.x5

2680.

sample

(-20%

1179.

w

(-40%,

29.13f2.91

25.4-1

2.5.ilf2.57

101.0

1’1, Fl

1920. 1088. sample

Q

(-60%

101.7

S9.91f5.99

24.30

24.32f2.43

i2.63fi.26

VI. VR

1919.

X4.53f8.45

1035. sample

23.06f2.31

v1

(-80%

D)

2680.

“1

39.26

1’3

1924

” Cross

sections approximate

’ The

calculated

in Table the

trace

in units center values sections

116.1

22.19

Q

IV, weighted cross

3x.99*3.90

113.1

952.

’ The

f10.17

60.70

0)

2685.

HDS

f14.0

30.x4

D)

2672.

HDS

139.x

136.1

VI, v3

1902.

sample

f14.3

21.14f2.11

D)

2669.

HDS

142.6

20.93

879. HDS

4.Xfktl.00

142.7

1927.

fll.61

21.96*2.20

of lOm”6 m2jsr. of gravity

of the

are the sums by the mole (Tables

fraction

I and

band,

given

of the anisotropy

for identification cross

for th? isotopic

sections species

purpows. for the

involved

as determined

bands,

in thr

given

analysis

of

II).

B. Anisotropy Spectra In a similar approach (28) to that used for the trace scattering, the absolute differential Raman anisotropy scattering cross section for the ith normal mode is a function of the derivative of the polarizability anisotropy for that mode, and is given in the SI system by (29)

(%);= (is)

(iXJ)[ 1 - exp( -hcw;/kT)]-‘g;y’?.

(3)

Here, since the rotational band contour of the anisotropy scattering can extend over a large frequency range, the fourth power of the wavenumber of the scattered radiation has been weighted by the rotational intensities and averaged. The invariant 7’: is the anisotropy of the derivative of the molecular polarizability tensor with respect to q; , gi is the degeneracy of the vibration, and all other quantities have been defined above. In terms of the irreducible spherical tensor components, the anisotropy of the polarizability is given by

436

FERNANDEZ-SANCHEZ

AND

MURPHY

The combinations of the irreducible spherical tensor polarizability related to the more familiar Cartesian components by (31) cy; = 6 -“2(2cX;, a$ - al,

= 2ia,

- (Y, - (YY_“)

CY:- al, = -2a,

components

are

Ly: + (u2, = Ly, - (YYJ aY: + (~1, = - 2icu,,,.

(5)

It can be more convenient to work in terms of these combinations of the irreducible spherical tensor components, since they usually transform according to the various symmetry species of the molecular point group. In this manner, the minimum number of independent intensity parameters can be obtained. The derivatives of these components with respect to normal coordinates are again expanded in terms of derivatives with respect to symmetry coordinates using a relation similar to Eq. (2); the latter derivatives are the intensity parameters for anisotropy scattering. In the I’ representation of H2S, which relates the Cartesian axes of the molecule to its inertial axes, the x-axis is the dipole axis and the y-axis is perpendicular to the molecular plane. In this axis system, the asymmetric stretching coordinate, S,, and the irreducible spherical polarizability tensor component a? - (Y2 1transform according to the Bi symmetry species of the C’,, molecular point group. The rotational corrections needed to properly compare the polarizability derivatives for the various isotopic species have been calculated with respect to a unit-mass reference molecule, as described previously (28). Initial estimates of the intensity parameters for the unit-mass reference molecule (28) are adjusted in a nonlinear optimization procedure to fit the observed anisotropy scattering cross sections. In each iteration of the calculation, the reference molecule parameters are transformed to those for each isotopic species. The contributions to a spectrum from several isotopic species may be taken into account, as in the trace scattering calculation, and, again, each cross section is given a weight proportional to the inverse of its estimated error. The error propagation technique (30) is used here, as well, to estimate the parameter error limits. IV. RESULTS

AND

DISCUSSION

A. Trace Scattering The observed Raman trace scattering cross sections for the recorded spectra are presented in Tables I and II, and compared with those calculated in the fit in terms of the dG /aSj intensity parameters using the relations given in Eqs. ( 1) and ( 2). The required normal coordinate matrix was obtained from the harmonic force field (32); the molecular geometry and the vibrational mode numbering scheme were also taken from that reference. The mole fraction results are also included in these tables. For the D2S sample they correspond to a deuterium abundance of 97 atom%, in agreement with the supplier’s specification. The trace scattering cross sections for each isotopic species are included in Table IV as calculated from the a& /as, values (Table V) which best fit the observed cross sections. The relative signs of the dG /&S, parameters are determined in the fit, and the sign of &/as, is chosen to be positive, in agreement with previous arguments

RAMAN INTENSITY OF HYDROGEN

SULFIDE

437

TABLE IV Calculated Differential Cross Sections for the Trace and Anisotropy Raman Scattering Spectra of Hydrogen Sulfide” Molecule and

Anisotropy

Trace

mode

au/a52

fRq”eIlCYb

ao/afi

frequencyb

HzS 4

2613.

295.8

‘2658.

44.55

uz

1183.

1.22

1220.

26.52

2658.

127.9

192i.

:37.91

Q DzS Vl

189s

250.2 1.37

X56.

uz ?J

Xi9.

20.i5

1927.

109.0

X6.46

HDS

” Rand

w3

2620.

145.1

2680.

VI

1902.

127.9

1902.

i3.02

u2

1034.

1.32

1060.

23.99

positions

b The

listed

frequency

in units

band

term

of cm-‘:

frequency

in Eqs.

cross

sections

is its center

(1) and

in 10e3”

of gravity.

m’jsr.

which

is used

to estimate

the

fourth

power

(3).

the theoretical calculations (22). The agreement between the observed and calculated trace cross sections is excellent, well within the reduced error limits, and indicates that these cross-section data for hydrogen sulfide satisfy the double-harmonic approximation very well. Note that, as found previously for a variety of molecules, the trace scattering cross sections for the bending modes are two orders of magnitude smaller than those for the stretching modes. Previously reported measurements of the total cross section of the uI Q branch for H2S (514.5 nm) include 270 X 1O-36 m2/sr (8) and 290 X 1O-36 m’/sr (7). These values are approximations to the trace cross section measured in this work: to the extent that the contribution of the anisotropy scattering to the total cross section (3.3) and

TABLE V Raman Scattering Intensity Parameters for Hydrogen Sulfide” Symmetry

coordinate,

5'1, symSH

&,

aalas,

S,

stretch

2.155iO.018

SH2 bending

SJ, asym

aa;jas, aco4 + nl,)ias,a(4 - nz,)ias,

0.107f0.003

1.013~0.016

2.OOOkO.032

(0.70,1.51)

(1.32,2.36)

0.60lf0.012

mo.555*o.01

(0.51,0.71)

(-0.71.-0.28)

1

m4.174f0.065b

SH stretch

(p4.08,-4.27)

o Units

of 10m30 CmV-‘.

For the

anisotropy

the extreme b The number -0.005.

isotopic

values

of the

variation

of decimal

The

error

parameters,

plaws

limits

constraints of a(ai given

are estimated

the quantities used $,)/a&, (see

text).

standard

in parentheses in the

numerical

for the

propagated

the range

procedure

due to the ‘rotational In addition,

deviations,

indicate

in the fit.

of results

found

for

is smaller

than

the

(see text). correction’,

HDS

species.

a(n:

112,)/a.%

is

438

FERNANDEZ-SANCHEZ

AND MURPHY

(found here to be about 7%) may be ignored, they are an upper limit to the trace cross section. With this caveat, their agreement with the present result to within the 10% error limit conventionally given for cross section measurements is felt to be satisfactory. In the spectra of the isotopic mixtures, the weak Q branches due to each of the three species are well resolved in the bending region. However, the corresponding bands in the SH and SD stretching regions are only partially resolved, and we have decided not to separate the contributions of the different isotopic species to the stretching region cross sections. This results in a numerical system where there is a high correlation ( -0.90) between the isotopic abundances in each of the mixtures; this in turn leads to the high estimated errors for these abundances (Table II). This situation may be contrasted with that in the acetylene study (28)) where the isotopic abundances were estimated with an order of magnitude better accuracy than found here. B. Anisotropy Scattering The measured anisotropy scattering cross sections are collected in Table III, and compared with those calculated in the fitting procedure. The mole fractions of the various isotopic species were fixed at the values determined in the trace scattering analysis. The analysis of the anisotropy spectrum is more complicated than that of the trace spectrum due to several reasons. In the first place, as was mentioned above, there are two independent anisotropy parameters for the totally symmetric modes, but only one cross-section datum, so that the two parameters cannot be determined directly in the cross-section analysis. Secondly, for H2S and D$, the symmetric vI and antisymmetric v3 stretching modes overlap in the spectrum so that only the combined cross section for the two modes can be determined experimentally. Thus, additional information must be used in order to determine each of the independent components of the derivatives of the molecular polarizability. For the totally symmetric bands of asymmetric top molecules, the intensities of vibration-rotation transitions depend on the matrix elements of the two irreducible spherical polarizability tensor components CY$ and CY~ + (~2, ( 13). Since these contributions can reinforce or interfere with each other depending on the rotational quantum numbers involved, the rotational profile of the band depends on the ratio of these components,

Note that R2,, depends on the axis system chosen for the molecule (I’, in this case), and that alternative versions of this ratio have been used in the literature. (A comparison of the various quantities has been given in Refs. (3) and ( I1 ) .) In some previous studies, the calculated rotational contour has been least-squares fit to the observed one, with R2,, as one of the fitting parameters. We have chosen not to do this in the present work; we feel that such an approach might be compromised by details such as the choice of broadening function for the calculated spectrum, and that a single sum-of-squares function might not respond appropriately to the subtle variations which occur on changing R 20. We have instead used the program described in Ref. ( 13) to calculate spectra for several well-separated values of RzO, and have selected the one which best resembles the observed one. Then, this calculated spectrum was subtracted from a second having a slightly different RzO value, and the strongest features in the difference spectrum were used to identify those in the calculated spectrum

RAMAN

INTENSITY

OF HYDROGEN

439

SULFIDE

which were most sensitive to changes in R 20. These features were then compared to their counterparts in the observed spectrum to select the calculated spectrum having the optimum value of R20. It turns out that often the spectral features which are most sensitive to changes in RzOare weaker and not well separated from nearby strong ones. It is difficult to see how such variations could be discerned in a lower resolution spectrum. Previous experimentally determined RzO values (converted to our definition, and in the I’ axis system) for the bending band, ~2, of HzS include -0.82 (9), -0.8 16 (IO), and -0.89 (II ), where the highest spectral resolution considered, 2 cm-‘, was that in Ref. ( 9). The value found in the ab initio calculation (22) is -0.8 16, while, for a pure bending coordinate, a value of -2/ fi( = -0.816) may be predicted (13) from the zeroth-order bond polarizability model. This latter relation was, in fact, taken as an assumption in several of the previous analyses. In Fig. 1, the spectrum of the v2 band of H$, measured with a spectral slit width of 0.5 cm-‘, is compared with a spectrum calculated with R20 = - 1.O. The energy levels and wavefunctions were calculated with molecular constants reported for the ground state ( 1.5) and the v2 excited state ( 17). Transitions were included for initial levels up to J = 20, the sample temperature was taken to be 300 K, and the calculated spectra were broadened with a Gaussian function with a width of 0.8 cm-’ . This width reasonably accounts for the broadening effect of the spectrometer resolution, the width of the laser excitation, and the smaller contribution from pressure broadening. Spectra calculated with RzO values of -0.5 and - 1.5 compared generally poorer with the observed one, for the features in the calculated spectrum sensitive to changes in RzO. However, as can be seen in Fig. 1, the comparison is at best semiquantitative.

3

1000

1100

1200

Wavenumber

1300

shift,

1400

1500

1600

cm”

FIG. 1. The ~2 bending band in the Raman spectrum of H2S. The experimental spectrum (top), observed with a OS-cm-’ spectral slit, is the total, unpolarized spectrum, so that trace scattering contributes in the region of the band origin, 1182.6 cm-i. The calculated spectrum (bottom) has been convoluted with a Gaussian broadening function with a half-width (FWHM) of 0.8 cm-‘. (See text for details of the calculation.)

440

FERNANDEZ-SANCHEZ

AND MURPHY

Some of the deviations could be due to noise in the observed spectrum, but we can make out similar differences when we compare lower resolution calculated spectra with their lower noise, experimental counterparts. It is clear that higher order contributions to the polarizability tensor expression should be considered. However, our Raman spectra are not at high enough resolution and do not have low enough noise to be treated on an empirical basis as were the absorption spectra (16). It will be necessary to consider the algebraic derivation of the various terms, as has been done for the pure rotational spectrum (23). In summary, we are presently choosing to report for the v2 band, RX, = - 1.O -+ 0.5, where the error is estimated from the variation of RX, required to give a noticeably poorer calculated spectrum when compared with the observed one. The SH stretching region of the H2S spectrum is complicated by the Coriolis interaction between vi and u3. In the previous consideration of this region (9), the observed spectrum was analysed on the basis of the sum of rigid-rotor spectra calculated separately for vi and u3. However, the matrix elements of (_yiand CY~ + (~2, for vl and ofa: - (~2, for v3 each connect a given ground state level with the same excited state level resulting from the Coriolis interaction of yI and u3. Since this interaction is expected to be significant in this case (the vl and v3 band origins (16) are at 2612.36 and 2626.12 cm-‘, respectively), and since the matrix elements may again interfere constructively or destructively, we have extended our spectral calculation program (13) to determine the combined vi and v3 transition matrix element which is then used to calculate the transition intensity. In the previous study of this band ( 9), where the experimental spectrum considered was measured with a resolution of 0.8 cm-‘, the relative anisotropies of v1and u3were found to be r’*(g1)/r’*(q3) = 0.361, and, for vl, a value of R20 = 1.13 was found. For comparison, the ab initio calculation (22) gives yf2(q1)/-yf2( 43) = 0.426, with the sign of a( a: - a?1)/dq3 opposite to those for d&dq, and ~(LY:+ a?2)/dq,, and RZJ = 1.462. (Note that the phase of q3 is here defined opposite to that given in Ref. (22) .) The zeroth-order bond polarizability model predicts (13) R20 = c*G/(2 - 3c*), where c is the cosine of the AB2 half-angle, for the symmetric stretching mode of an AB2 molecule having Cz, symmetry. This gives R20 = 2.1 1 for v1 of HS. In Fig. 2, the spectrum of the SH stretching region of H$, measured with a spectral slit width of 0.5 cm-‘, is compared with the spectrum calculated for -y’2(q1)/ y’*(q3) = 0.348, Rzo = 2.0, and the signs of the polarizability components as found in the ab initio calculation (22). The energy levels and wavefunctions were calculated with molecular constants reported for the ground state (1.5) and the interacting vI and ~3 excited states ( 16). Transitions were included for initial levels up to J = 20, and the calculated spectra were broadened with a Gaussian function with a width of 0.8 cm-‘. The procedure used to determine the best calculated spectrum was similar to that used for the v2 band. Since the v3 intensity contribution is significantly larger than that of vI , R20( q, ) was fixed at 1.5, corresponding to the ab initio prediction, and the relative anisotropy, T’~(q, )/y’*( q3), was varied. The best value was found to be 0.348, with significantly poorer comparisons for spectra calculated with 0.290 and 0.426. (In the actual calculation, the relative values of the polarizability derivatives with respect to ql and q3 were constrained; however, these constraints are expressed more clearly in terms of the relative anisotropy, which leads to the odd values given here.) The intensities of the weak “side bands” in the region 2450-2500 cm-’ were especially sensitive to changes in this ratio. The relative anisotropy was then fixed at this value, and the Rzo( q, ) value was varied, holding y’2 (q, ) fixed. Here, because of the smaller

RAMAN

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SULFIDE

I

I

I

I

I

I

2400

2500

2600

2700

2800

2900

Wavenumber

10

shift, cm-’

FIG. 2. The SH stretching region of the Raman spectrum of H$. The experimental spectrum (top), observed with a OS-cm-’ spectral slit, is the perpendicularly polarized component, equivalent to the anisotropy spectrum. (Note that there is a trace scattering contribution in the region of the band origin, 26 12.4 cm-’ .) The calculated spectrum (bottom) has been convoluted with a Gaussian broadening function with a halfwidth (FWHM) of 0.8 cm-‘. (See text for details of the calculation.)

contribution of v1 to the total band intensity, it was barely possible to discern differences in the calculated spectra for RX, values of 1.5 and 2.5. We thus choose the nominal value Rlo(ql) = 2.0 + 1.0. Finally, we calculated the spectrum for L?(a: - a?,)/dq, having the opposite sign; there were changes in the calculated spectrum of up to 1OY” of the maximum intensity, and the comparison of this spectrum with the observed one is obviously poorer. In the calculation of these spectra, we verified numerically that the total intensity was independent of the signs of the polarizability components, i.e., that the expression for the anisotropy (Eq. (4), above) is not affected by the Coriolis interaction between vl and v3. Here, the comparison between the best calculated spectrum and the observed one is much better than found for the u2 spectrum. There are differences, tentatively ascribed to higher order contributions, but these are relatively minor compared to those for the bending band. Thus we find r’2(ql)/y’2(q3) = 0.35 + 0.06, &(ql) = 2.0 +- 1.0, and the relative signs of the polarizability components agree with those found in the ab initio calculation (22). The above results provide the constraints needed for a stable numerical system in the fit of the anisotropy cross sections. These constraints cannot be applied directly, since these relations are for the components of the polarizability derivatives with respect to the normal coordinates of H2S, while the parameters in the cross-section fitting procedure are the derivatives with respect to symmetry coordinates. Since the S1 and S2 symmetry coordinates are mixed to some extent in the totally symmetric normal coordinates q, and q2, it is necessary to adjust the constraints on the symmetry coordinate derivatives to give normal coordinate derivatives which are related in the

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desired manner. This was done by manual iteration. The necessary relations were found to be RZO(Sl) = 1.973, &(&) = -0.924, and the ratio of a(a: - a?r)/d& to a&as, is -4.112. Using these results, we have determined all of the allowed polarizability component derivatives; they are given in Table V. Because there are relatively large error limits for the constraint relations used in the fit, we have also repeated the fit for each constraint at its limiting values. The range of results so obtained for each of the intensity parameters is also included in Table V. Due to the relatively large errors in the RzO constraints, the ranges for the Sr and S, anisotropy parameters are much larger than the errors propagated in the fit. Only for the S, parameter is the range of variation consistent with the error estimate. As mentioned above, the anisotropy cross-section calculation takes into account the rotational corrections to the symmetry coordinate derivatives, which depend on the components of the equilibrium polarizability. However, since the equilibrium second-rank components are very small for hydrogen sulfide 1, )/Xi', are negligible (0.01%) (3, 20), the resulting isotopic variations of a(& - CY compared to the estimated error; we here report (Table V) a single value for this parameter. The relative signs of the parameters are determined in the fit; their absolute signs have been chosen to agree with the theoretical predictions (22). The polarizability derivatives with respect to normal coordinates, obtained from this analysis, are presented in Table II of the accompanying article (23), where they are used to predict the effect of centrifugal distortion on the rotational Raman spectrum of hydrogen sulfide. Again for the anisotropy scattering, the agreement between the observed and calculated cross sections is very good; for this case also, the assumptions of the doubleharmonic approximation appear to be closely followed. V. CONCLUSION

We have measured absolute Raman trace and anisotropy scattering cross sections for the vibrational spectra of hydrogen- and deuterium-substituted isotopic species of hydrogen sulfide. These results have been analyzed in terms of derivatives of polarizability tensor components with respect to symmetry coordinates. The constraints required for the anisotropy parameters were obtained by modeling the rotational structure in the vibrational bands of H&L For both the trace and anisotropy cross sections, the observed values can be fit well within the estimated experimental errors, which demonstrates that the double-harmonic approximation is well fulfilled in these cases. On the whole, our results confirm the previous experimental and theoretical determinations of these quantities. Thus, it appears that our concerns about the earlier experimental results were largely misplaced. The advantage of the current results is that they provide a self-consistent set of absolute intensity parameters which can be used with some confidence in further studies of this molecule, for example, in the consideration of the centrifugal distortion contribution to the pure rotational Raman spectrum presented in the accompanying article (23). ACKNOWLEDGMENT JMF thanks Ministerio de Educaci6n y Ciencia of Spain for a postdoctoral fellowship supported by the FPI Program. RECEIVED:

April

1,

1992

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