Computing Phonons Within Density Functional Theory: Finite Difference vs. Density Functional Perturbation Theory

Computing phonons within density functional theory (DFT) is typically achieved using finite difference or density functional perturbation theory (DFPT). Both of these approaches will yield the numerically exact solutions if judiciously executed, but converging either method can be challenging in crystals with sensitive phonons. Here we use the irreducible derivative finite difference approach for computing phonons and compare to DFPT in three simple systems where the phonons can be challenging to converge: Al, NaCl, and cubic AuZn. All calculations are performed using the same DFT code, allowing for a one-to-one comparison. The irreducible derivative approach is used to compute phonons in two distinct ways, using either second-order energy derivatives or first-order force derivatives, which are then compared to DFPT. The convergence of the results are studied as a function of the number of electronic k-points, plane-wave cutoff, and the electronic self-consistent convergence tolerance, demonstrating the rate of convergence in each case. The overall performance of each method is compared, giving practical guidelines on the relative computational cost when computing non-trivial phonons, such as the charge density wave in AuZn.