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Molybdenum purple bronze Li$_{0.9}$Mo$_{6}$O$_{17}$ is an exceptionalmaterial known to exhibit one dimensional (1D) properties for energies down toa few meV. This fact seems to be well established both in experiments and inband structure theory. We use the unusual, very 1-dimensional band dispersionobtained in \emph{ab-initio} DFT-LMTO band calculations as our starting pointto study the physics emerging below 300meV. A dispersion perpendicular to themain dispersive direction is obtained and investigated in detail. Based onthis, we derive an effective low energy theory within the Tomonaga Luttingerliquid (TLL) framework. We estimate the strength of the possible interactionsand from this deduce the values of the TLL parameters for charge modes. Finallywe investigate possible instabilities of TLL by deriving renormalization group(RG) equations which allow us to predict the size of potential gaps in thespectrum. While $2k_F$ instabilities strongly suppress each other, the $4k_F$instabilities cooperate, which paves the way for a possible CDW at the lowestenergies. The aim of this work is to understand the experimental findings, inparticular the ones which are certainly lying within the 1D regime. We discussthe validity of our 1D approach and further perspectives for the lower energyphases.

Luttinger liquid theory of purple bronze $Li_{0.9}Mo_6O_{17}$ in the charge regime