A 'moment-conserving' reformulation of GW theory

We show how to construct an effective Hamiltonian whose dimension scales linearly with system size, and whose eigenvalues systematically approximate the excitation energies of $GW$ theory. This is achieved by rigorously expanding the self-energy in order to exactly conserve a desired number of frequency-independent moments of the self-energy dynamics. Recasting $GW$ in this way admits a low-scaling $\mathcal{O}[N^4]$ approach to build and solve this Hamiltonian, with a proposal to reduce this further to $\mathcal{O}[N^3]$. This relies on exposing a novel recursive framework for the density response moments of the random phase approximation (RPA), where the efficient calculation of its starting point mirrors the low-scaling approaches to compute RPA correlation energies. The frequency integration of $GW$ which distinguishes so many different $GW$ variants can be performed without approximation directly in this moment representation. Furthermore, the solution to the Dyson equation can be performed exactly, avoiding analytic continuation, diagonal approximations or iterative solutions to the quasiparticle equation, with the full-frequency spectrum obtained from the complete solution of this effective static Hamiltonian. We show how this approach converges rapidly with respect to the order of the conserved self-energy moments, and is applied across the $GW100$ benchmark dataset to obtain accurate $GW$ spectra in comparison to traditional implementations. We also show the ability to systematically converge all-electron full-frequency spectra and high-energy features beyond frontier excitations, as well as avoiding discontinuities in the spectrum which afflict many other $GW$ approaches.