We investigate the emergence of chaotic dynamics in a quantum Fermi -Pasta -Ulam problem for anharmonic vibrations in atomic chains applying semi-quantitative analysis of resonant interactions complemented by exact diagonalization numerical studies. The crossover energy separating chaotic high energy phase and localized (integrable) low energy phase is estimated. It decreases inversely proportionally to the number of atoms until approaching the quantum regime where this dependence saturates. The chaotic behavior appears at lower energies in systems with free or fixed ends boundary conditions compared to periodic systems. The applications of the theory to realistic molecules are discussed.determines the localization threshold. The threshold energy can be redefined in terms of the critical temperature corresponding to that energy.The knowledge of the localization threshold for an individual molecule is significant since the vibrational relaxation changes dramatically depending on whether the energy of the molecule is lower or higher than the threshold [5,10,32,33]. In the latter case the vibrational relaxation follows standard Fermi Golden rule kinetics [34], while in the localized regime it is much slower. Therefore, the present work is focused on the localization threshold and its dependence on system size (number of atoms) and the strength of anharmonic interaction.Since the properties of a molecule can be sensitive to its shape the consideration is restricted to the simple linear chain of atoms coupled by anharmonic interactions identical to the FPU problem [25]. This problem is relevant for the energy relaxation and transport in polymer chains used in the modern heat conducting devices [3,32,35,36]. The anomalous increase of a thermal conductivity there with the system size suggests a very slow thermalization or even the lack of one [36]. The results for the FPU problem can be qualitatively relevant for the analysis of more complicated molecules.The consideration is restricted to quantum mechanical systems. It has been suggested that the threshold energy separating localized and chaotic states decreases with the system size [27][28][29][37][38][39][40]. This leads to the reduction of thermal energy below the vibrational quantization energy, which makes quantum effects inevitably significant for sufficiently large molecules.The paper is organized as follows. The FPU problems with different boundary conditions are formulated and briefly discussed in Sec. 1. The analysis of localization is performed combining analytical (Sec. 2) and numerical (Sec. 3) approaches for the FPU problems with different boundary conditions. Both approaches are reasonably consistent with each other and led to the predictions of analytical dependencies of localization threshold on system parameters that are discussed in Sec. 4 for organic molecules. The methods and brief conclusions are formulated in Secs. 5, 6.