Electron-hole response function of transition metal trichalcogenides NbSe$_3$ and monoclinic-TaS$_3$

NbSe$_3$ and monoclinic-TaS$_3$ ($m$-TaS$_3$) are quasi-1D metals containing three different types of chains and undergoing two different charge density wave (CDW) Peierls transitions at T$_{P_1}$ and T$_{P_2}$. The nature of these transitions is discussed on the basis of first-principles DFT calculation of their electron-hole Lindhard response function. As a result of stronger inter-chain interactions the Fermi surface (FS) and Lindhard function of NbSe$_3$ are considerably more complex than those for $m$-TaS$_3$; however a common scenario can be put forward to rationalize the results. The intra-chain inter-band nesting processes dominate the strongest response for both type I and type III chains of the two compounds. Two well-defined maxima of the Lindhard response for NbSe$_3$ are found with the (0$a$*, 0$c$*) and (1/2$a$*, 1/2$c$*) transverse components at T$_{P_1}$ and T$_{P_2}$, respectively, whereas the second maximum is not observed for $m$-TaS$_3$ at T$_{P2}$. Analysis of the different inter-chain coupling mechanisms leads to the conclusion that FS nesting effects are only relevant to set the transverse $a$* components in NbSe$_3$. For the transverse coupling along $c$* in NbSe$_3$ and along both $a$* and $c$* for $m$-TaS$_3$, one must take into account the strongest inter-chain Coulomb coupling mechanism. Phonon spectrum calculations show the formation of a giant 2$k_F$ Kohn anomaly in $m$-TaS$_3$. All these results support the weak coupling scenario for the Peierls transition of transition metal trichalcogenides.