FEAST nonlinear eigenvalue algorithm for $GW$ quasiparticle equations

The use of Green's function in quantum many-body theory often leads to nonlinear eigenvalue problems, as Green's function needs to be defined in energy domain. The $GW$ approximation method is one of the typical examples. In this article, we introduce a method based on the FEAST eigenvalue algorithm for accurately solving the nonlinear eigenvalue $G_0W_0$ quasiparticle equation, eliminating the need for the Kohn-Sham wavefunction approximation. Based on the contour integral method for nonlinear eigenvalue problem, the energy (eigenvalue) domain is extended to complex plane. Hypercomplex number is introduced to the contour deformation calculation of $GW$ self-energy to carry imaginary parts of both Green's functions and FEAST quadrature nodes. Calculation results for various molecules are presented and compared with a more conventional graphical solution approximation method. It is confirmed that the Highest Occupied Molecular Orbital (HOMO) from the Kohn-Sham equation is very close to that of $GW$, while the Least Unoccupied Molecular Orbital (LUMO) shows noticeable differences.