To obtain accurate enthalpies, we added zero-point energies (ZPE) and phonon vibrational energies (U vib ) to the DFT-derived electronic energies. ZPE and U vib were calculated under the independent harmonic oscillator approximation as implemented in the ASE Thermochemistry class. We found that the combined vibrational energies are significant for protonated (H x MnO 2 • yH 2 O) structures (Fig. ). In fact, a linear relationship exists between ZPE+U vib and the number of hydroxyl groups in H x MnO 2 • yH 2 O compounds. For ion-inserted structures without hydroxyl groups (A x MnO 2 , A ≠ H), we approximated the combined vibrational energies as that of pure b-MnO 2 .