Computational Chemistry

wavelength rangeDl. The total area of all the rectangles is the total flux emitted
over its whole wavelength range by the blackbody. AsDlapproaches zero (note
that for the nonmonochromatic radiation from a blackbody the fluxata particular
wavelength is essentially zero) the histogram approaches being a smooth curve, the
ratio of finite increments approximates a derivative, and we can ask: what is the
function (Fig.4.2)dF/dl¼f(l)? In the answer to this question lay the beginnings of
quantum theory.
Late nineteenth century physics, classical physics at its zenith, predicted that the
flux density emitted by a blackbody should rise without limit as the wavelength
decreases. This is because classical physics held that radiation of a particular fre-
quency was emitted by oscillators (atoms or whatever) vibrating with that frequency,
and that the average energy of an oscillator was independent of its frequency; since the
number of possible frequencies increases without limit, the flux density (energy per
second per unit area per wavelength interval) from the blackbody should rise without
limit toward higher frequencies or shorter wavelengths, into the ultraviolet, and so the
total flux (energy per second per unit area) should be infinite. This is clearly absurd and
was recognized as being absurd; in fact, it was called “the ultraviolet catastrophe” [ 1 ].
To understand the nature of blackbody radiation and to escape the ultraviolet catas-
trophe, physicists in the 1890s tried to find the function (Fig.4.2)f(l).
Without breaking with classical physics, Wien had found a theoretical equation
that fit the Lummer–Pringsheim curve at relatively short wavelengths, and Rayleigh
and Jeans one that fit at relatively long wavelengths. Max Planck^2 adopted a
different approach: he found, in 1900, a purely empirical equationdF/dl¼f(l)

FACT

∆F

∆λ

∆F

∆λ

CLASSICAL PHYSICS

(FALSE)

λλ

Fig. 4.2In the limit the bar graph becomes a curve, the graph off(l) versusl, where
fðlÞ¼lim
Dl! 0

DF
Dl¼
dF
dl, essentially intensity of radiation versus wavelength. Planck’s efforts to find
the functionf(l) led to the quantum theory

(^2) Max Planck, born Kiel, Germany, 1858. Ph.D. Berlin 1879. Professor, Kiel, Berlin. Nobel Prize in
physics for quantum theory of blackbody radiation 1918. Died G€ottingen, 1947.
88 4 Introduction to Quantum Mechanics in Computational Chemistry