The emission of cold dust grains at long wavelengths will soon be observed by the Planck and Herschel satellites and will provide new constraints on the nature of interstellar dust. In particular, the microwave galactic anomalous foreground detected between 10 to 90GHz, proposed as coming from small spinning grains (PAHs), should help to define these species better. Moreover, understanding the fluctuations of the anomalous foreground quantitatively over the sky is crucial for CMB studies. We focus on the long-wavelength emission of interstellar PAHs in their vibrational and rotational transitions. We present here the first model that coherently describes the PAH emission from the near-IR to microwave range. We take quantum effects into account to describe the rotation of PAHs and compare our results to current models of spinning dust to assess the validity of the classical treatment used. Between absorptions of stellar photons, we followed the rovibrational radiative cascade of PAHs. We used the exact-statistical method of Draine & Li (2001ApJ...551..807D
) to derive the distribution of PAH internal energy and followed a quantum approach for the rotational excitation induced by vibrational (IR) transitions. We also examined the influence of the vibrational relaxation scheme and of the low-energy cross-section on the PAH emission. We study the emissivity of spinning PAHs in a variety of physical conditions (radiation field intensity and gas density), search for specific signatures in this emission that can be looked for observationally, and discuss how the anomalous foreground may constrain the PAH size distribution. Simultaneously predicting the vibrational and rotational emission of PAHs, our model can explain the observed emission of the Perseus molecular cloud from the IR to the microwave range with plausible PAH properties. We show that for λ≥3mm the PAH vibrational emission no longer scales with the radiation field intensity (G0
), unlike the mid-IR part of the spectrum (which scales with G0
). This emission represents less than 10% of the total dust emission at 100GHz. Similarly, we find the broadband emissivity of spinning PAHs per carbon atom to be rather constant for G0
≤100 and for proton densities nH
. In the diffuse ISM, photon exchange and gas-grain interactions play comparable roles in exciting the rotation of PAHs, and the emissivity of spinning PAHs is dominated by the contribution of small species (bearing less than 100 C atoms). We show that the classical description of rotation used in previous works is a good approximation and that unknowns in the vibrational relaxation scheme and low-energy cross-section affect the PAH rotational emissivity around 30GHz by less than 15%. The contrasted behaviour of the PAH vibrational and rotational emissivities with G0
provides a clear prediction that can be tested against observations of anomalous and dust mid-IR emissions: this is the subject of a companion paper. Comparison of these emissions complemented with radio observations (21cm or continuum) will provide constraints on the fraction of small species and the electric dipole moment of interstellar PAHs.