Calculation of phonons in real-space density functional theory

We present an accurate and efficient formulation for the calculation of phonons in real-space Kohn-Sham density functional theory. Specifically, employing a local exchange-correlation functional, norm-conserving pseudopotential in the Kleinman-Bylander representation, and local form for the electrostatics, we derive expressions for the dynamical matrix and associated Sternheimer equation that are particularly amenable to the real-space finite-difference method, within the framework of density functional perturbation theory. In particular, the formulation is applicable to insulating as well as metallic systems of any dimensionality, enabling the efficient and accurate treatment of semi-infinite and bulk systems alike, for both orthogonal and nonorthogonal cells. We also develop an implementation of the proposed formulation within the high-order finite-difference method. Through representative examples, we verify the accuracy of the computed phonon dispersion curves and density of states, demonstrating excellent agreement with established plane-wave results.

Figure 1

Phonon dispersion and density of states for bcc cesium, $\alpha$-phosphorene, and polyyne. The red (continuous) and the blue (dashed) curves in the dispersion plots represent the cubic spline fit to the data points obtained from abinit and m-sparc, respectively. The maximum differences in the phonon frequencies between m-sparc and abinit for the cesium, phosphorene, and polyyne systems are 0.2, 0.93, and $0.61{\mathrm{cm}}^{-1}$, respectively.