  Interatomic pairwise methods are currently among the most popular and
accurate ways to include dispersion energy in density functional theory (DFT)
calculations. However, when applied to more than two atoms, these methods are
still frequently perceived to be based on \textit{ad hoc} assumptions, rather
than a rigorous derivation from quantum mechanics. Starting from the adiabatic
connection fluctuation-dissipation (ACFD) theorem, an exact expression for the
electronic exchange-correlation energy, we demonstrate that the pairwise
interatomic dispersion energy for an arbitrary collection of isotropic
polarizable dipoles emerges from the second-order expansion of the ACFD
formula. Moreover, for a system of quantum harmonic oscillators coupled through
a dipole--dipole potential, we prove the equivalence between the full
interaction energy obtained from the Hamiltonian diagonalization and the ACFD
correlation energy in the random-phase approximation. This property makes the
Hamiltonian diagonalization an efficient method for the calculation of the
many-body dispersion energy. In addition, we show that the switching function
used to damp the dispersion interaction at short distances arises from a
short-range screened Coulomb potential, whose role is to account for the
spatial spread of the individual atomic dipole moments. By using the ACFD
formula we gain a deeper understanding of the approximations made in the
interatomic pairwise approaches, providing a powerful formalism for further
development of accurate and efficient methods for the calculation of the
dispersion energy.
