  Eigenstate thermalization has been shown to occur for few-body observables in
a wide range of nonintegrable interacting models. For intensive observables
that are sums of local operators, because of their polynomially vanishing
Hilbert-Schmidt norm, weak eigenstate thermalization occurs in quadratic and
integrable interacting systems. Here, we unveil a novel weak eigenstate
thermalization phenomenon that occurs in quadratic models whose single-particle
sector exhibits quantum chaos (quantum-chaotic quadratic models) and in
integrable interacting models. In such models, we show that there are few-body
observables with a nonvanishing Hilbert-Schmidt norm that are guarrantied to
exhibit a polynomially vanishing variance of the diagonal matrix elements, a
phenomenon we dub normal weak eigenstate thermalization. For quantum-chaotic
quadratic Hamiltonians, we prove that normal weak eigenstate thermalization is
a consequence of single-particle eigenstate thermalization, i.e., it can be
viewed as a manifestation of quantum chaos at the single-particle level. We
report numerical evidence of normal weak eigenstate thermalization for
quantum-chaotic quadratic models such as the 3D Anderson model in the
delocalized regime and the power-law random banded matrix model, as well as for
the integrable interacting spin-$\frac{1}{2}$ XYZ and XXZ models.
