Solving Nonlinear GW Quasiparticle Eigenvalue Equations with a Contour Integral-Based Algorithm

Green's function methods in quantum many-body theory frequently give rise to nonlinear eigenvalue problems, as Green's functions are defined in the energy domain. The $GW$ approximation is a prominent example. In this paper, we present a method based on the FEAST eigenvalue algorithm for accurately solving the nonlinear eigenvalue problem of the $G_0W_0$ quasiparticle equation, bypassing the need for the Kohn-Sham wavefunction approximation and the spectrum method. Using the contour integral method for nonlinear eigenvalue problems, the energy (eigenvalue) domain is extended to the complex plane. We introduce hypercomplex numbers to perform contour deformation calculations in the $GW$ self-energy, capturing the imaginary components of both Green's functions and FEAST quadrature nodes. Results from calculations on various molecules are presented and compared to a more conventional graphical solution approximation and spectrum method. Our findings show that the Highest Occupied Molecular Orbital (HOMO) from the Kohn-Sham equation is closely aligned with that from $GW$, while the Lowest Unoccupied Molecular Orbital (LUMO) exhibits more significant differences.