## 3 Results
Using a harmonic oscillator model [27], \(\omega=(\varphi/M)^{1/2}\), where \(M\) represents the mass of the guest, the estimated effective force constant \(\varphi\) of Cs is 0.85 eV A\({}^{-1}\) in Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and 0.96 eV A\({}^{-1}\) in Cs\({}_{8}\)Si\({}_{128}\). The values of \(\varphi\) for Rb modes in Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (see Figure 1) and Rb\({}_{8}\)Si\({}_{128}\) are calculated to be 0.31 and 0.46 eV A\({}^{-1}\), respectively. Furthermore, an estimation on the Al-Si bond length and Si-Si bond length are based on the nearest neighboring distance table from our LDA calculation. The average Al-Si bond length is computed to be 2.42 A, while the average Si-Si bond length is 2.32 A. This difference has been identified to be relevant with a structural change upon the partial framework substitution. The physical origin of the lattice structure response upon the framework substitution might be connected to the altered guest-host interaction strength due to the Al-Si formation. In Figure 1, one of the notable features for low-frequency (<200 cm\({}^{-1}\)) vibration spectrum is the nondispersive and flat optic modes, whose frequencies remain at nearly 41.1 cm\({}^{-1}\) and 31.3 cm\({}^{-1}\) for Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\), respectively. Accordingly, the determined vibrational density of states (VDOS) denoted by \(g(\omega)\) is shown in Figure 2, displaying the acoustic phonon bandwidth (<115 cm\({}^{-1}\)) along with the rattling optical modes (Cs, Rb mode). \(g(\omega)\) is normalized such that \(\int g(\omega)\ \mathrm{d}\omega=3N\), where \(N\) is the total number of atoms in the primitive unit cell when the frequency \(\omega\) is limited to the entire phonon region (<500 cm\({}^{-1}\)). The [PERSON] approximation which assumes that acoustic phonon branches have linear dispersion relation within first Brillouin zone is taken into account to calculate phonon density of states in Figure 2. Meanwhile, the projected density of states is obtained by summation of that contribution over flat, localized optical phonon bands due to guest rattler (Cs, Rb) in the same diagram. On the other hand, the group velocity for the lowest-lying acoustic phonons in these ternary compounds are almost independent of the guest type, remaining unaffected when compared with Al\({}_{8}\)Si\({}_{128}\) in Figure 3.  
Numerical calculation involving diagonalization of the dynamical matrix verifies that these localized optic bands are essentially due to the guest motion. Comparing these low-lying optic modes to that of the dispersion relation of Al\({}_{8}\)Si\({}_{128}\) in Figure 3, one sees the \"bending\" of the acoustic branches, in which the frequency is supposed to extend to approximately 120 cm\({}^{-1}\). This is indicative of a strong coupling between the localized Cs (or Rb) modes and acoustic phonon bands, which carry thermal energy. Therefore, these resonant couplings are identified as \"avoided-crossing effect\", allowing phonon branches of the Cs (or Rb) and Al\({}_{8}\)Si\({}_{128}\) framework to meet and avoid one another near 41 cm\({}^{-1}\) and 31 cm\({}^{-1}\) in the Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) material, respectively. The lattice vibration contribution to the thermal conductivity in some semiconducting clathrate compounds were found to play a dominant role in comparison with the relatively small electronic  
Figure 3: Predicted low-frequency vibration spectrum of Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\).Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) and Al\({}_{8}\)Si\({}_{128}\). Figure 2: Calculated vibrational density of states (VDOS) vs. frequency for Cs\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\) (solid line) and Rb\({}_{8}\)Al\({}_{8}\)Si\({}_{128}\)(dotted line). contribution [28; 29; 30].