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A first-principles-based effective Hamiltonian is developed and used, along with direct ab-initio techniques, to investigate finite-temperature properties of the system commonly coined the most complex perovskite, that is NaNbO3. Such simulations successfully reproduce the existence of seven different phases in its phase diagram. The decomposition of the total energy of this effective Hamiltonian into different terms, altogether with the values of the parameters associated with these terms, also allow to shed some lights into puzzling features of such compound. Examples include revealing the microscopic reasons of why R3c is its ground state, and why it solely adopts in-phase tiltings at high temperatures versus complex nanotwins for intermediate temperatures. The results of the computations also call for a revisiting of the so-called P , R and S states, in the sense that an unexpected and previously overlooked inhomogeneous electrical polarization is numerically found in the P state while novel complex tiltings associated with the simultaneous condensation of several k-points are predicted for the controversial R and S phases.

Understanding and revisiting the most complex perovskite system via atomistic simulations