  We present Einstein coefficient spectra and a detailed-balance derivation of
generalized Einstein relations between them that is based on the connection
between spontaneous and stimulated emission. If two broadened levels or bands
overlap in energy, transitions between them need not be purely absorptive or
emissive. Consequently, spontaneous emission can occur in both transition
directions, and four Einstein coefficient spectra replace the three Einstein
coefficients for a line. At equilibrium, the four different spectra obey five
pairwise relationships and one lineshape generates all four. These
relationships are independent of molecular quantum statistics and predict the
Stokes' shift between forward and reverse transitions required by equilibrium
with blackbody radiation. For Boltzmann statistics, the relative strengths of
forward and reverse transitions depend on the formal chemical potential
difference between the initial and final bands, which becomes the standard
chemical potential difference for ideal solutes. The formal chemical potential
of a band replaces both the energy and degeneracy of a quantum level. Like the
energies of quantum levels, the formal chemical potentials of bands obey the
Rydberg-Ritz combination principle. Each stimulated Einstein coefficient
spectrum gives a frequency-dependent transition cross section. Transition cross
sections obey causality and a detailed-balance condition with spontaneous
emission, but do not directly obey generalized Einstein relations. Even with an
energetic width much less than the photon energy, an absorptive forward
transition with an energetic width much greater than the thermal energy can
have such an extreme Stokes' shift that its reverse transition cross section
becomes predominantly absorptive rather than emissive.
