## 3 Results  
Ternary Clathrate Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\)  
The LDA-calculated lattice constant, bulk modulus and cohesive energy per atom in the presence of Al-substituted and Ga-substituted binary, ternary compounds are listed in Table 1. Here, Ga appearing in (Cs,Rb)\({}_{8}\)GaSi\({}_{128}\) has an s\({}^{2}\)p\({}^{1}\) valence electronic configuration, making it possible for it to behave as an electron acceptor while compensating for excess valence electrons that arise from guest impurity according to the Zintle-Klemm criteria [7]. It is noted that all Al atoms subjected to their occupied 8\(a\) Wyckoff sites can form an Al-Si bond in the case of (Rb,Cs)\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\). Intriguingly, our density functional theory (DFT) work shows the entire lattice structure experiences a slight expansion when all Si atoms at the 8\(a\) Wyckoff sites are completely replaced by Al. From the viewpoint of fundamental chemistry, aluminum is classified as a Group-III element, which has a reduced atomic mass (approximately 4% lighter than Si) and a smaller radius. In contrast, the lattice constant of Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (14.676 A) is computed to remain approximately 0.64% larger than that of Cs\({}_{8}\)Si\({}_{136}\) (14.583 A). This discrepancy in terms of the structural change allows one to anticipate the correlation between the framework bonding formation Al-Si and guest-host coupling. Our results give rise to approximately 41.1 cm\({}^{-1}\) for Cs vibration frequency \(\omega\) in Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (see Figure 1), which is approximately 5.9% lower than that of Cs\({}_{8}\)Si\({}_{136}\). The suppressed Cs vibration frequency in the Al-substituted compound Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) is an indicative of its relatively weakly bound behavior with respect to cage constituents. In other words, the Cs guest has slightly more room to move around inside the cage, resulting in an occurrence of geometry dilation. Our results in Table 1, which describe the structural property of Rb-containing ternary compounds, yield a similar lattice constant: 14.670 A for Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and 14.644 A for Rb\({}_{8}\)Ga\({}_{8}\)Si\({}_{128}\). Analogous to the expanded volume found in Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) compared with Cs\({}_{8}\)Si\({}_{136}\), lattice framework expansion also occurs in Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) when all Si atoms at the 8\(a\) Wyckoff sites are replaced by Al atoms in Rb\({}_{8}\)Si\({}_{136}\). The rattling modes of Rb in Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) show frequencies of 31.3 cm\({}^{-1}\), which are approximately 17.4% smaller than the value of the guest vibration in Rb\({}_{8}\)Si\({}_{136}\). Therefore, it is further expected that the formation of the Al-Si bond other than the Si-Si bond might participate in minimizing the effective force constant of the Rb rattler, leading to a suppressed Rb vibration. \begin{table}
\begin{tabular}{c c c c c} \hline
**Material** & \(\mathbf{a}\) **(Å)** & \(\mathbf{E_{0}}\) **(eV/atom)** & **K (Gpa)** & **dK/dP** \\ \hline Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) & 14.669 & \(-\)5.555 & 78.428 & 4.172 \\ Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) & 14.676 & \(-\)5.573 & 80.385 & 3.082 \\ Rb\({}_{8}\)Ga\({}_{8}\)Si\({}_{128}\) & 14.644 & \(-\)5.536 & 79.063 & 5.054 \\ Cs\({}_{8}\)Ga\({}_{8}\)Si\({}_{128}\) & 14.651 & \(-\)5.555 & 79.735 & 4.085 \\ Cs\({}_{8}\)Si\({}_{136}\) & 14.583 & \(-\)5.694 & 82.783 & 5.205 \\ Rb\({}_{8}\)Si\({}_{136}\) & 14.572 & \(-\)5.681 & 83.115 & 4.955 \\ \hline \end{tabular}
\end{table} 
Table 1: The equation of state (EOS) parameters obtained for the filled ternary clathrate (Rb,Cs)\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\), (Rb,Cs)\({}_{8}\)Ga\({}_{8}\)Si\({}_{128}\) and binary clathrate (Cs,Rb)\({}_{8}\)Si\({}_{136}\).