  Kagome Materials with flat bands exhibit wildly different physical properties
depending on symmetry group, and electron number. For the case of ScV$_6$Sn$_6$
in space group 191, we investigate the existence of a charge density wave (CDW)
at vector $\bar{K}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ and its relationship
with the phonon behavior. The experimental findings reveal a $\sim$95K CDW
without nesting/peaks in the electron susceptibility at $\bar{K}$. Notably,
ScV$_6$Sn$_6$ exhibits a collapsed phonon mode at
$H=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$ and an imaginary flat phonon band in
the vicinity of $H$. The soft phonon is attributed to triangular Sn ($Sn^T$)
mirror-even vibrations along the $z$-direction. We develop a simple force
constant model to describe the entire soft phonon dispersion. By employing a
new (Gaussian) approximation of the hopping parameter, we demonstrate the
renormalization of the phonon frequency and the consequent collapse of the $H$
phonon. Additionally, we propose an effective model with two order parameters
(OPs) to explain the appearance of the CDW at $\bar{K}$, which competes with
the collapsed phonon at $H$. Through comparisons with experimental data, we
show that the $H$ OP undergoes a second-order phase transition while exhibiting
substantial fluctuations, ultimately inducing the first-order transition of the
$\bar{K}$ OP. Furthermore, we extend our analysis to the similar compound
YV$_6$Sn$_6$, which lacks a CDW phase, attributing this difference to the
participation of the heavier Y atom in the out-of-plane phonon behavior. Our
comprehensive study not only elucidates the CDW in ScV$_6$Sn$_6$ but also
presents a significant advancement in modeling complex electronic systems,
fostering collaborations between ab-initio simulations and analytical
approaches.
