Efficient dielectric matrix calculations using the Lanczos algorithm for fast many-body $G_0{W}_0$ implementations

We present a $G_0{W}_0$ implementation that assesses the two major bottlenecks of traditional plane-waves implementations, the summations over conduction states and the inversion of the dielectric matrix, without introducing new approximations in the formalism. The first bottleneck is circumvented by converting the summations into Sternheimer equations. Then, the novel avenue of expressing the dielectric matrix in a Lanczos basis is developed, which reduces the matrix size by orders of magnitude while being computationally efficient. We also develop a model dielectric operator that allows us to further reduce the size of the dielectric matrix without accuracy loss. Furthermore, we develop a scheme that reduces the numerical cost of the contour deformation technique to the level of the lightest plasmon pole model. Finally, the use of the simplified quasiminimal residual scheme in replacement of the conjugate gradients algorithm allows a direct evaluation of the $G_0{W}_0$ corrections at the desired real frequencies, without need for analytical continuation. The performance of the resulting $G_0{W}_0$ implementation is demonstrated by comparison with a traditional plane-waves implementation, which reveals a 500-fold speedup for the silane molecule. Finally, the accuracy of our $G_0{W}_0$ implementation is demonstrated by comparison with other $G_0{W}_0$ calculations and experimental results.

Figure 1

The path used in the contour deformation technique. The poles of the screened Coulomb interaction ${\widehat{W}}_0\left(\omega \right)$ lie outside the contour, only some poles of the Green's function ${\widehat{G}}_0\left(\omega \right)$ lie inside. This figure is reproduced with permission from Ref. [33].

Figure 2

Convergence study on the expectation value of the correlation part of the self-energy for the HOMO orbital of the silane molecule ${\Sigma }_e^{\mathrm{c}}\left(0\right)$ using the present and the conventional $G_0{W}_0$ implementation of the abinit project. The convergence studies are carried out with respect to the size of the dielectric matrix, which is controlled by the number of Lanczos vectors $N_L$ in the first implementation and the number of plane waves used to describe the dielectric matrix $N_{PW\epsilon }$ in the second. The horizontal lines show the energy zone considered to be converged and the vertical lines show the approximate CPU time required to reach this level of convergence, based on a linear interpolation between the first data point to be converged and the preceding one.