  Nonlinear optics, and particularly second harmonic generation (SHG), is
increasingly used in many modern disciplines from material characterization in
physical sciences to bio-imaging in medicine and optical signal processing in
information technology. We present a theoretical analysis providing a strong
estimate for the energy-integrated SHG response. Requirements of symmetry in
time and space are fully respected in the calculation, and estimates of natural
and magnetic circular dichroic signals are superior to previous ones. Like
symmetry requirements are traced in the amplitude for magnetic neutron
scattering, which includes all axial and polar (Dirac) contributions. Our
method of working, in terms of spherical multipoles and implementation of
symmetry, could be of use in a variety of other probes of electronic
properties.
