  In the framework of the second-quantization representation of the master
equation governing the irreversible epitaxial growth, exact equations
describing the evolution of the island densities has been obtained. Their
decoupling within a mean field-type approximation with the unknown correlation
functions replaced by capture numbers (CNs) has been used to derive a closed
set of rate equations. The latter has been compared with the exact equations to
obtain rigorous definitions of the CNs. The CN that describes the nucleation of
dimer islands from two mobile monomers has been measured in the exact kinetic
Monte Carlo simulations with the use of the rigorous definition. Strong
disagreement with the literature values calculated within alternative
techniques has been found, especially at low surface coverage. Plausible causes
for the discrepancies are suggested.
  Another important result of the rigorous approach is the rate equation with
the term describing the monomer diffusion. The equation significantly differs
from the widely used equations of this kind known from literature. Arguments in
favour of our approach are given.
