  Raman spectra obtained by the inelastic scattering of light by crystalline
solids contain contributions from first-order vibrational processes (e.g. the
emission or absorption of one phonon, a quantum of vibration) as well as
higher-order processes with at least two phonons being involved. At second
order, coupling with the entire phonon spectrum induces a response that may
strongly depend on the excitation energy, and reflects complex processes more
difficult to interpret. In particular, excitons (i.e. bound electron-hole
pairs) may enhance the absorption and emission of light, and couple strongly
with phonons in resonance conditions. We design and implement a
first-principles methodology to compute second-order Raman scattering,
incorporating dielectric responses and phonon eigenstates obtained from
density-functional theory and many-body theory. We demonstrate our approach for
the case of silicon, relating frequency-dependent relative Raman intensities,
that are in excellent agreement with experiment, to different vibrations and
regions of the Brillouin zone. We show that exciton-phonon coupling, computed
from first principles, indeed strongly affect the spectrum in resonance
conditions. The ability to analyze second-order Raman spectra thus provides
direct insight into this interaction.
