## 1 Introduction  
Ternary and quaternary clathrate compounds have been the subject of growing interest in recent years [1; 2; 3; 4; 5; 6]. Here, the terms \"ternary\" and \"quaternary\" refer to a chemical substance that contains three and four distinct elements, respectively. Generally, clathrate compounds are classified as one of two categories: Type-I and Type-II. The unit cell of a Type-I compound configuration possesses a simple cubic framework that accommodates a limited integer number of guest atoms placed inside the polyhedron structure, which is a 20-atom or 24-atom cage. Similarly, the framework of a Type-II compound is made of 20- and 28-atom cages connected in a ratio of 2:1, leading to encapsulation of the guest atoms. Moreover, the clathrate framework serving as the host can be formulated by one Group-IV element (Si, Ge, Sn) or a mixture of any single type of Group-IV element and another type of Group-III atoms (usually Al or Ga). Specifically, the usage of Al or Ga atoms is selected to compensate for excess valence electrons arising from guest atoms in order to make the entire compound semiconducting. That is, the Zintl phase criteria [7] is satisfied after such Group-III atoms are substituted onto the framework site in the presence of alkali or alkali earth guests. Another unique characteristic of the clathrate compound structure is the flexibility of filling two different guest atom types rather than a single type into two different sized cages. Previously, ternary and quaternary clathrate compounds,including Sr\({}_{8}\)Ga\({}_{x}\)Si\({}_{46\text{-}x}\), X\({}_{8}\)Ga\({}_{16}\)Ge\({}_{30}\) (X = Eu, Sr, Ba), Ba\({}_{8}\)Zn\({}_{x}\)Ge\({}_{46\text{-}x-y}\)Si\({}_{y}\), and (Ba,Sr)\({}_{8}\)Ga\({}_{16}\)Si\({}_{x}\)Ge\({}_{30\text{-}x}\), have been undergoing intensive study [8; 9; 10; 11; 12; 13]. In contrast to investigations on pure clathrate, such as open framework Si\({}_{46}\), Ge\({}_{46}\), Si\({}_{136}\), or intermetallic binary clathrate systems Ba\({}_{x}\)Si\({}_{46}\), A\({}_{x}\)Si\({}_{136}\) (A = alkaline metal; \(0<x\leq 24\)) [14; 15; 16; 17; 18], little is understood about Type-II ternary and quaternary clathrate compounds as opposed to Type-I counterparts. Therefore, in this work we decided to intensively study Si-based Type-II materials R\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (R = Cs, Rb) and Cs\({}_{8}\)Na\({}_{16}\)Al\({}_{24}\)Si\({}_{112}\). These materials are of great importance due to their unique thermodynamical properties. Specifically, the existence of low-energy, localized optic modes arising from guest \"rattlers\" cause the acoustic phonon spectrum to exhibit an \"avoided-crossing\" effect. Therefore, the suppression of acoustic phonon bands might have an inherent interaction with increased anharmonicity in these semiconducting systems. Consequently, a reevaluation of the phonon lifetime is needed because the three-phonon process begins to dominate over other scattering models derived from a harmonic approximation point of view. In this paper, significant details will also be discussed on the underestimation of the lattice thermal conductivity when considering the neighboring distance of guest rattlers as the phonon mean free path. In other words, the primitive model, which suggests that guest atoms act as scattering centers, needs to be readjusted because of the large phonon thermal conductivity observed in Cs\({}_{8}\)Na\({}_{16}\)Si\({}_{136}\). In addition, we studied the impact of different substitutional framework atoms on the thermodynamic natures of these semiconducting clathrates (e.g., Cs\({}_{8}\)Ga\({}_{8}\)Si\({}_{128}\) and Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\)) of interest. We found an acoustic band difference exists for lattice vibration when comparing the compound R\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (R = Cs, Rb) to its \"parent\" Al\({}_{8}\)Si\({}_{128}\) framework.