Lattice dynamics and electron-phonon coupling calculations using non-diagonal supercells

We study the direct calculation of total energy derivatives for lattice dynamics and electron-phonon coupling calculations using supercell matrices with non-zero off-diagonal elements. We show that it is possible to determine the response of a periodic system to a perturbation characterized by a wave vector with reduced fractional coordinates $(m_1/n_1,m_2/n_2,m_3/n_3)$ using a supercell containing a number of primitive cells equal to the least common multiple of $n_1$, $n_2$, and $n_3$. If only diagonal supercell matrices are used, a supercell containing $n_1n_2n_3$ primitive cells is required. We demonstrate that the use of non-diagonal supercells significantly reduces the computational cost of obtaining converged zero-point energies and phonon dispersions for diamond and graphite. We also perform electron-phonon coupling calculations using the direct method to sample the vibrational Brillouin zone with grids of unprecedented size, which enables us to investigate the convergence of the zero-point renormalization to the thermal and optical band gaps of diamond.