Physical Chemistry Third Edition

970 23 Optical Spectroscopy and Photochemistry

basis functioneiωt. The frequency of oscillation is

ν

ω

2 π

(23.3-15)

The functionc(ω) plays the same role as the expansion coefficients in a linear combi-

nation and expresses the intensities of the different frequencies. It is called theFourier

transformofI(t).^10 It contains the same information asI(t), encoded as a function of

frequency instead of time. It is the infrared spectrum of the sample.

To find the spectrumc(ω), one must invert the Fourier transform, which is done by

calculating the integral

c(ω)

1

√

2 π

∫∞

−∞

I(t)e−iωtdt (23.3-16)

Since Eq. (23.3-14) and Eq. (23.3-16) differ only in the sign of the exponent,I(t)is

also called the Fourier transform ofc(ω).

EXAMPLE23.8

Find the Fourier transform of the functionf(ω)1/a^2 +ω^2 , whereais a positive constant.

Solution

I(t)

1

√

2 π

∞∫

−∞

eiωt

a^2 +ω^2

dω

1

√

2 π

∞∫

−∞

cos(ωt)+isin(ωt)

a^2 +ω^2

dω

The real part of the integrand is an even function, and the imaginary part is an odd function.

The imaginary part will vanish upon integration, and the even part will give twice the value

of the integrand from 0 to∞:

I(t)

2

√

2 π

∞∫

−∞

cos(ωt)

a^2 +ω^2

dω

2

√

2 π

π

2 a

e−a|t|

√

π

2

1

a

e−a|t|

where we have looked the integral up in a table.^11

The Fourier transform of the interferogram must be carried out numerically. Fourier

transform infrared (FTIR) instruments have a dedicated computer to do this. A numer-

ical procedure known as thefast Fourier transform(FFT) is generally used.^12 The

procedure is carried out repeatedly for different frequencies. Depending on the instru-

ment, the intensity might be obtained for values of the reciprocal wavelength differing

by1cm−^1 or 2 cm−^1. With 1 cm−^1 resolution (spacing), a spectrum from 500 cm−^1

to 4000 cm−^1 would consist of 3501 separate points representing the percent trans-

mittance or the absorbance at each reciprocal wavelength. A graph of the spectrum

(^10) See L. Glasser,J. Chem. Educ., 64 , A261 (1987); andJ. Chem. Educ., 64 , A306 (1987) for an introduction
to Fourier series and transforms.
(^11) H. B. Dwight,Tables of Integrals and Other Mathematical Data, 4th ed., Macmillan, New York, 1961.
(^12) J. R. Barrante,Applied Mathematics for Physical Chemistry, 3rd ed., Prentice Hall, Upper Saddle River,
NJ, 2004, p. 182ff.