Effect of Alkaline Metal Filling on the Structural Properties of Type-II Clathrate A$_{\mathrm{x}}$M$_{136} $(A $=$ Na,K,Rb,Cs; M $=$ Group IV Atom; 0$\le $ x$\le $ 24)

Early investigations of the properties of the Type II clathrate Na$_{\mathrm{x}}$Si$_{136}$ (0\textless x\textless 24) [1] have found that, as the composition x increases, the Si$_{136}$ lattice exhibits framework contraction upon filling (0\textless x\textless 8), followed by an expansion of the unit cell volume (9\textless x\textless 24). Stimulated by this discovery of a non-monotonic structural response to cage filling by the guests, we have performed a systematic, first-principles study the of the guest-framework interaction in the large and small cages in the Type II clathrates Si$_{136}$, Ge$_{136}$, and Sn$_{136}$. Our calculations are based on the VASP code and we have considered Na, K, Rb, and Cs guests for 0$\le $x$\le $24. An emphasis in our study is on how the guest atom size affects the dynamical behavior of the host material. We focus on the host lattice structural expansion or compression as x increases. We also present and discuss calculations of the effective potential energy curves for the guest-host interactions in these materials. Our results are correlated with the harmonic (or anhamonic) oscillations of the guests. These results are useful as an indication of the expected behavior of the guest ``rattling'' phonon modes in these materials. Among other results, we find that some guests are weakly bonded in the host cages and others are unstable around the cage centers. \\[4pt] [1] Beekman, Nenghabi, Biswas, Myles,~ Baitinger, Grin, Nolas,~Inorganic Chemistry~49,~5338~(2010).