Test Results

Figure 6: Left: Free energy vs. smearing factor $k_B T$. The inset shows a zoomed in part of the graph, comparing different values of free energy for different Methfessel-Paxton order of approximations. Right: The difference between free energy and the calculated ground-state energy (called Harris energy here) vs. smearing factor $k_B T$.

Figure 6 shows the comparison between calculations on a 4 atom unit cell aluminium bulk using Fermi and Methfessel-Paxton smearing methods. We only did non-self-consistency calculations. We can see that Fermi smearing leads to different energy results as we increase smearing factor $k_B T$. We can see that the difference between Free energy and the calculated ground-state energy (we called it Harris energy) also increased dramatically for Fermi smearing. This is expected as the smearing does correspond to the physical temperature and as $k_B T$ increases we depart from the zero-temperature regime. On the other hand, the smearing in Methfessel-Paxton method is just a parameter used in the approximation. And we can see that increasing $k_B T$ has small effect on the energies calculated using Methfessel-Paxton smearing and results also improve as the order of Hermite polynomials used increases. This means a relatively large smearing factor can be used under Methfessel-Paxton approximation, which allows fewer $\vec{k}$-points. To be more specific, figure 7 shows that for smearing factor $k_B T = 0.1\text{Ha} = 2.72\text{eV}$, (corresponding to smearing temperature of about 31565.51 K), we reach $\vec{k}$-point convergence at about 10 Bloch space ($\vec{k}$) points per each reciprocal lattice direction, while the total energy will be within $10^{-5}$Ha of the ground state energies calculated using low smearing and large number of $\vec{k}$-points. This is compared with the requirement of 25 $\vec{k}$-points for convergence with a small smearing factor, see figure 8.

Figure 7: $\vec{k}$-point convergence for aluminium bulk, 4 atoms cell, with order 5 Methfessel-Paxton smearing.

Figure 8: LEFT: Convergence of the ground-state energy and free energy with increased real space grid finess. RIGHT: Convergence of the ground-state energy and free energy with increased number of $\vec{k}$-points in all directions.