Test of Lattice Constant with Correction from Zero-Point Energy

In Born-Oppenheimer system, the total energy doesn't include the kinetic energy of the nucleus. The zero point energy of the crystal can influence the lattice constant found by minimizing the total energy. The zero point anharmonic expansion (ZPAE) can be estimated in the Debye model. We can also use a direct way to get the zero point energy by calculating the phonon frequency using density-functional perturbation theory. Some solids were tested using different functionals to get the total energy and using DFPT to get the zero-point energy. We also expanded the vibration frequency as the compression ratio, which can give us the trends of the zero-point energy. For those different correction ways, the phonon frequency correction should be the most precise method in theory. The Debye model gives a reasonable approximation in most of those solids, but for Diamond structure and the Zinc-Blende structures, the Debye model may overestimate the correction. The expansion frequency way also overestimates the corrections compared to the phonon frequency correction.