ThisSemiclassical theory chapter introduces the semi-classical theory of radiation. The theory is semi-classical in the sense that, the radiation is described by Maxwell equations, while the matter which is absorbing and emitting the radiation, is treated quantum mechanically. The theory provides an explanation for the vast majority of spectroscopic and electromagnetic phenomena that are encountered in chemistry. This chapter builds on the time-dependent perturbation theory previously encountered in Chap. 11 . The density matrixDensity matrix is introduced and used to calculate the real and imaginary parts of the molecular polarizability, a function which was previously encountered in its classical form in Chap. 7 . Afterwards, the Kramers–Kronig relationsKramers – Kronig relations are derived. This pair of equations serves to link the real part of the polarizability to the imaginary part and vice versa.