  Numerically optimized reduced descriptions of macromolecular liquids often
present thermodynamic inconsistency with atomistic level descriptions even if
the total correlation function, i.e. the structure, appears to be in agreement.
An analytical expression for the effective potential between a pair of
coarse-grained units is derived starting from the first-principles
Ornstein-Zernike equation, for a polymer liquid where each chain is represented
as a collection of interpenetrating blobs, with a variable number of blobs,
$n_b$, of size $N_b$. The potential is characterized by a long tail, slowly
decaying with characteristic scaling exponent of $N_b^{1/4}$. This general
result applies to any coarse-grained model of polymer melts with units larger
than the persistence length, highlighting the importance of the long,
repulsive, potential tail for the model to correctly predict both structural
and thermodynamic properties of the macromolecular liquid.
