  Fullerene solids doped with alkali metals (A$_3$C$_{60}$, A = K, Rb, Cs)
exhibit a superconducting transition temperature ($T_c$) as high as 40 K, and
their unconventional superconducting properties have been a subject of debate.
With application of high pressure on K$_3$C$_{60}$ and Rb$_3$C$_{60}$, the
experiments demonstrate the decrease of $T_c$. In this paper, we focus on
K$_3$C$_{60}$ and derive the structure of K$_3$C$_{60}$ under different
pressures based on first-principles calculations, exploring the trends of
Coulomb interactions at various pressures. By utilizing the Maximally Localized
Wannier function approach, Constrained Density Functional Perturbation Theory
(cDFPT), and Constrained Random Phase Approximation (cRPA), we construct a
microscopic low-energy model near the Fermi level. Our results strongly
indicate that, in the K$_3$C$_{60}$ system, as pressure increases, the effect
of phonons is the key to intraorbital electron pairing. There is a dominance of
the phonon-driven superconducting mechanism at high pressure.
