Interplay of H$_{2}$O and K$^+$ inside the channels of Mn$_{8}$O$_{16}$

With the rapid growth in consumer electronics and electric vehicles, there is an increasing interest in developing high-density batteries, which requires investigation of robust electrode materials. One of these, $\alpha$-MnO$_{2}$, is inexpensive and environmentally benign to manufacture. It consists of an arrangement of corner- and edge- shared MnO$_{6}$ octahedra forming a $2 \times 2$ tunnel structure, and belongs to a family of ``octahedral molecular sieve structures" (OMS-$2$). Owing to the large tunnel cavity of OMS-$2$, cations such as K$^{+}$, Li$^{+}$, Ag$^{+}$, etc. as well as water molecules can be introduced into the $2\times2$ tunnel, thereby enabling us to tailor its chemical and physical properties. In this work, we focus on the incorporation of K${^+}$ in the tunnel, which stabilizes $\alpha$-MnO$_{2}$, in agreement with experiment. Our primary goal is to investigate the role of water in stabilizing the ions already present in a tunnel cavity, using first-principles density functional theory (DFT) calculations, including van der Waals interactions. We also analyze how the hydrogen-bond network competes with the ionic bonding of K$^+$ in the channel.