The equation of motion for a harmonically bound classical electron interacting with an electric field
is given by the Drude–Lorentz equation
is the natural frequency of the oscillator and
is the damping constant. For an incident electromagnetic field of frequency
at the point
can be conveniently represented by a complex exponential
. The steady-state solution, in complex form, is given by
. The electric dipole moment of the electron,
, corresponds to the macroscopic relation for the polarizability
is the complex electric susceptibility
being the number of polarizable electrons per unit volume. It is sufficient to approximate
, with transition frequencies lying in the optical region
. The real part of the susceptibility
gives the frequency dependence (dispersion) of the dielectric constant
and index of refraction
. The imaginary part
represents the absorption coefficient. This function has the form of a Lorentzian.
In the more accurate quantum theory of dispersion, the frequency
is replaced by a sum over several atomic transition frequencies and the damping parameters
are determined by excited-state lifetimes.
The real and imaginary parts of the susceptibility are connected by the Kramers–Kronig relations:
signifies the Cauchy principal value of the integral.
 L. Rosenfeld, Theory of Electrons
, New York: Dover Publications, 1965, pp. 68 ff.