## 4 Conclusions  
We report a systematic first-principles investigation on the structural and vibrational properties of R\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (R = Cs, Rb) and Cs\({}_{8}\)Na\({}_{16}\)Al\({}_{24}\)Si\({}_{112}\) clathrates. Among them, vibration of the rattler in Al-substituted ternary clathrate R\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) produces lower rattling modes than that of the same rattler in R\({}_{8}\)Si\({}_{128}\). Therefore, it is anticipated that the formation of Al-Si might participate in a minimization of the guest rattling frequency in (Cs,Rb)\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\), compared to that of (Cs,Rb)\({}_{8}\)Si\({}_{136}\). Moreover, the validity of the resonant rattling model where the acoustic phonon can travel between  
Figure 5: Theoretically and experimentally estimated isotropic atomic displacement parameters (\(U_{\mathrm{iso}}\)) of Cs in the 28-cage. The open symbols represent the data corresponding to Cs vibration in Cs\({}_{8}\)Na\({}_{16}\)Ge\({}_{136}\)[22]. The solid symbol denotes the measured value of \(U_{\mathrm{iso}}\) regarding Cs rattling in Cs\({}_{8}\)Na\({}_{16}\)Al\({}_{24}\)Si\({}_{112}\).  
Figure 6: The predicted lattice thermal conductivity as a function of Al occupancy on the 96g Wyckoff site. The dotted line is used for comparison.  
the neighboring scattering center to the exchange thermal energy is reconsidered for materials studied here, since this scheme induces underestimation on predicted lattice thermal conductivity. Instead, computing the phonon lifetime governed by three-phonon processes near the crossing in R\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and Cs\({}_{88}\)Na\({}_{16}\)Al\({}_{24}\)Si\({}_{112}\) is preferred. In addition to this, the \"avoided-crossing\" effect causes no appreciable difference when evaluating the acoustic phonon speed to exist. The optimized geometry for Cs\({}_{88}\)Na\({}_{16}\)Al\({}_{24}\)Si\({}_{112}\) is determined theoretically when Al/Si ratio has values of 0.2/0.8 for the 8\(a\) Wyckoff sites, 0.1/0.9 for the 32\(e\) sites and 0.2/0.8 for the 96\(g\) sites, according to the suggested refinement from the experimental point of view. The calculated atomic displacement parameters (ADPs) of Cs for every possible composite structure of Cs\({}_{88}\)Na\({}_{16}\)Al\({}_{24}\)Si\({}_{112}\) are larger than the XRD-determined result at \(T=300\) K. This overestimation is thought to be attributable to the temperature-dependent quartic anharmonicity effect of guest vibration.  
[PERSON], the first author, undertook all of the (sometimes computationally intense) calculations that are reported in the paper. He also wrote the majority of the first draft and participated in the editing of that draft which produced the final, submitted version. [PERSON], the second author, is a Professor of Physics and Astronomy at Texas Tech University. He is the PhD research advisor to [PERSON].  
This research received no external funding.  
We would like to acknowledge grateful discussions with [PERSON] (Texas Tech University) about the use of VASP. We also appreciate many hours of computing time at the High Performance Computing Center of Texas Tech University.  
The authors declare no conflict of interest.