ILIs. I f Is were increased to equal IL, NF(2-photon) would increase to 5 × 107 photons/pulse but NF(stimulated Raman) would increase to 2 × 107 photons/pulse. Addi- tionally, the sample temperature in these experiments was limited by the 10 K temperature of the cryogenic refrigerator and the thermal contact between the two. Direct immersion in liquid helium would decrease the vibrational energy relaxation rate. Nevertheless, the two- photon contribution will remain a strong interference which should prevent analytical applications of this method.

The second method for performing double resonance using infrared excitation of the vibration should not suffer from the same interference. The two-photon proc- ess can still contribute to the fluorescence, but the effi- ciency of direct infrared excitation makes the double resonance process more than 2 orders of magnitude larger than the two-photon contribution. It is difficult, however, to generate infrared light at frequencies needed for the characteristic vibrations of interest and further work depends upon development of a suitable infrared source.

ACKNOWLEDGMENT

This research was supported by the National Science Foundation under Grants CHE7825306 and CHE8119893.

1. A. Seilmeier, W. Kaiser, and A. Laubereau, Opt. Commun. 26, 441 (1978). 2. A. Seilmeier, W. Kaiser, A. Laubereau, and S. F. Fischer, Chem. Phys. Lett.

58, 225 (1978). 3. J. C. Wright, Appl. Spectrosc. 34, 151 {1980). 4. I. V. Aleksandmv, Ya. S. Bobovich, V. G. Maslov, and A. N. Sidorov, Opt.

Spektrosk. 35, 264 (1973). 5. R. M. Hochstrasser and C. A. Nyi, J. Chem. Phys. 72, 2591 (1980). 6. A. Laubereau, G. Wochner, and W. Kaiser, Phys. Rev. A 13, 2212 (1976).

Measurement of Frequency Shifts Us ing Infrared or Raman Dif ference Spectroscopy

J A A N L A A N E

Department of Chemistry, Texas A&M University, College Station, Texas 77843

Index Headings: R a m a n s p e c t r o s c o p y .

We have recently developed a comprehensive picture of the difference spectra expected from the use of Raman difference spectroscopy (RDS). 1'2 Both frequency shifts and bandwidth changes contribute to the nature of such spectra. We have also described a number of experimen- tal applications where the RDS technique is used to measure frequency shifts and bandwidth changes with great precision. 3-6 This work as well as that from other laboratories has recently been reviewed. 7 In these publi- cations, it has been mentioned that the RDS theory is

Received 13 October 1982; revision received 6 December 1982.

474 Volume 37, Number 5, 1983

also immediately applicable to analyzing infrared absorp- tion bands; however, no experimental results were pre- sented. One reason for this is that it is inherently more difficult to obtain useful information on frequency shifts and bandwidth changes from infrared difference spectra due to the limitations on infrared absorbances. That is, it is generally impractical to work with infrared bands whose absorbances considerably exceed about 1.0 absor- bance unit (less than 10% transmittance), whereas for RDS large intensities are desirable for RDS measure- ments. Since many strong infrared bands go "off-scale" for pure liquids in even short path length cells, it is not practical to use infrared difference techniques to measure solvent shifts relative to the pure liquid for many sam- ples. However, frequency differences between dilute so- lutions may be measured.

Brown et al. s have recently analyzed the infrared dif- ference spectra arising from frequency shifts. These shifts may be anomalous, arising from errors in frequency reproducibility on repetitive scans. This is unlike the situation for the RDS method where two or four spectra are recorded simultaneously. For infrared spectra smaller reproducibility errors can be expected for Fourier trans- form instruments as compared to grating spectrometers. Brown et al. derived various relationships resulting from frequency differences between Gaussian bands and also presented several comparisons between predicted and experimental data. To a large extent these results agree with ours derived previously for RDS spectra. 1'2 How- ever, since the theory developed for RDS is considerably more detailed and covers both Gaussian and Lorentzian bands and also both frequency shifts and bandwidth changes, it is appropriate to reiterate here what the results are and how they are also applicable to infrared difference techniques.

First, we examine the basic line shape expressions. For a Lorentzian band

IL( ) = I o ( r / 2 ) (~ _ ~oo) 2 + (F/2)2 (1)

where IL(~o) is the band intensity (Raman intensity or infrared absorbance), which varies as a function of fre- quency z, Io is the maximum intensity at the band center ~o, and F is the band width measured at half-height (i.e., IL(wo _+ F / 2 ) = / 0 / 2 ) . Similarly, for a Gaussian band

Ig(~o) = Ioexp[-C(~o-~o)2/r 2] (2)

where C = 4 In 2 = 2.7726. For comparison purposes it is useful to note that Brown et al. use the definitions B and v instead of F and ~0 and also make use of

= (8 In 2)-v~r = (2c) -v~r = B/2.354. (3)

Thus, instead of Eq. (2) they have as their Eq. (4):

A(v) = A(o) exp (--v2/2a 2) (4)

where A@) represents the absorbance as a function of the frequency v, which has its origin at the band maxi- mum. In order to arrive at an expression for the infrared difference spectrum, Brown et al. defined a Gaussian shaped band in the transmission domain (arbitrarily setting the band maximum to have an absorbance of 1.0, or a transmittance of 0.1),

APPLIED SPECTROSCOPY

T(~) = 1 - 0.9 exp (-v2/2a2), (5)

and then proceeded to use

A(v) = - log T(D (6)

to define the absorbance as a function of frequency. It should be noted here that the a in Eq. (5) has been

redefined and is no longer equivalent to that used in Eq. (3). The new a (in the transmission domain) is given by

a = 8 In 1 -~ lO-'/;]J r = 0.6745r. (7)

Since most modern infrared instruments are capable of obtaining aborbance spectra either by computational or electronic conversion from transmittance spectra, it is not necessary to derive the characteristics of difference spectra in the transmission domain. In fact, the cumber- some derivations and results obtained for transmission spectra argue for the analyses to be done in the intensity (absorbance) domain. As a result, instead of using Eqs. (5) to (7), we prefer to retain the definitions of Eqs. (2) to (4).

Starting from Eq. (1) for Lorentzian bands and from Eq. (2) for Gaussian bands, the derivations for the dif- ference spectra (Raman or infrared) have been presented 1'2 and will not be duplicated here. Only the results will be summarized. For most cases the frequency shift is sufficiently small (4 < ~0.2 F) that accurate approximation formulas may be used. For Lorentzian bands the frequency shift can be determined from

A = (4/27) '/2 F d/ Io = 0.385 rd/I0 (8)

where d is the peak to peak height in the derivative-like difference curve and where I0 is the intensity of the original infrared or Raman band. It is assumed that both bands used for the spectral subtraction have the same intensity I0. The result of Eq. (8) is equivalent to Eq. (8) of Laane and Kiefer 1 and to Eq. (6) of Hirschfeld2 For Gaussian bands

= (8C)-V2e'~rd/Io = 0 . 3 5 0 r d / I o . (9)

Both Eqs. (8) and (9) are approximations which hold at low A/F ratios. When A/F > ~0.2 it is appropriate to use Table I of Ref. 1 to relate experimentally measured ratios of d/ Io to A/F. This approach is applicable whether

the frequency shift arises from physical reasons (solvent shifts, isotope shifts, etc.) or from instrumental artifacts. It can also be used for band shapes intermediate between Lorentzian and Gaussian types. Since the coefficients 0.385 and 0.350 in Eqs. (8) and (9) only differ by 10%, a rough approximation of Lorentzian character followed by interpolation may be used for mixed band types. More detail may be found elsewhere. 1'2

For small frequency shifts the maximum and minimum of a difference spectrum resulting from a frequency shift have been shown to occur at

~ M L I L00 ± 12-'tar = OJo ± 0.289F (10)

for a Lorentzian band and at

~ M G = LO0 ± (2C)-V2F = w0 ± 0.425F (11)

for a Gaussian band. Significantly, the position of the minimum and maximum do not change for small 5 / F ratios. For larger A/F values Table I of Ref. 1 lists 5/F ratios as a function of A/F (5 represents the frequency separation between the minimum and maximum in the difference spectrum).

In addition to frequency shifts, bandwidth charges can also be determined accurately using either infrared dif- ference or Raman difference spectroscopy. The theory and examples of this have been described previously. 2

In summary, Eqs. (8) and (9) of this paper may be used to calculate the frequency shift from difference spectrum. Although the theory was previously derived primarily for application to Raman spectroscopy, it is just as applicable to infrared absorbance spectra, whether the frequency shifts are real or instrumental artifacts.

ACKNOWLEDGMENT

This work was supported by the National Science Foundation.

1. J. Laane and W. Kiefer, J. Chem. Phys. 72, 5304 (1980). 2. J. Laane, J. Chem. Phys. 75, 2539 {1981). 3. J. Laane and W. Kiefer, J. Chem. Phys. 73, 4971 (1980). 4. J. Laane and W. Kiefer, Appl. Speetrosc. 35,267 (1981). 5. J. Laane and W. Kiefer, Appl. Spectrosc. 35,428 (1981). 6. J. Laane, H. Eichele, H. P. Hohenberger, and W. Kiefer, J. Mol. Speetrosc.

86, 262 (1981). 7. J. Laane, Vibrational Spectra and Structure, J. R. Durig, Ed. (1983), Vol. 12,

p. 405. 8. C. W. Brown, P. F. Lynch, and R. J. Obremski, Appl. Spectrosc. 36, 539

(1982). 9. T. Hirschfeld, Appl. Spectrosc. 30, 550 (1976).

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