Context : 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 90 GHz , 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 .

Aims : 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 .

Methods : 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 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 .

Results : 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 \lambda \geq 3 mm the PAH vibrational emission no longer scales with the radiation field intensity ( G _ { 0 } ) , unlike the mid-IR part of the spectrum ( which scales with G _ { 0 } ) .
This emission represents less than 10 % of the total dust emission at 100 GHz .
Similarly , we find the broadband emissivity of spinning PAHs per carbon atom to be rather constant for G _ { 0 } \leq 100 and for proton densities n _ { H } < 100 cm ^ { -3 } .
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 30 GHz by less than 15 % .

Conclusions : The contrasted behaviour of the PAH vibrational and rotational emissivities with G _ { 0 } 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 ( 21 cm or continuum ) will provide constraints on the fraction of small species and the electric dipole moment of interstellar PAHs .