  Quantum spin-1/2 antiferromagnetic Heisenberg trimerized chain with strong
intradimer and weak monomer-dimer coupling constants is studied using the novel
many-body perturbation expansion, which is developed from the exactly solved
spin-1/2 Ising-Heisenberg diamond chain preserving correlations between all
interacting spins of the trimerized chain unlike the standard perturbation
scheme developed on the grounds of noninteracting spin monomers and dimers. The
Heisenberg trimerized chain shows the intermediate one-third plateau, which was
also observed in the magnetization curve of the polymeric compound
Cu$_3$(P$_2$O$_6$OH)$_2$ affording its experimental realization. Within the
modified strong-coupling method we have obtained the effective Hamiltonians for
the magnetic-field range from zero to one-third plateau, and from one-third
plateau to the saturation magnetization. The second-order perturbation theory
shows reliable results even for the moderate ratio between weaker dimer-monomer
and stronger intradimer coupling constants. We have also examined thermodynamic
properties and recovered the low-temperature peak in the specific heat. The
accuracy of the developed method is tested through a comparison with numerical
density-matrix renormalization group and quantum Monte Carlo simulations. Using
the results for the effective Hamiltonian we suggest straightforward procedure
for finding the microscopic parameters of one-dimensional trimerized magnetic
compounds with strong intradimer and weak monomer-dimer couplings. We found the
refined values for the coupling constants of Cu$_3$(P$_2$O$_6$OH)$_2$ by
matching the theoretical results with the available experimental data for the
magnetization and magnetic susceptibility in a wide range of temperatures and
magnetic fields.
