Taming convergence in the determinant approach for x-ray excitation spectra

A determinant formalism in combination with ab initio calculations was proposed recently and has paved a way for simulating and interpreting x-ray excitation spectra in condensed-phase systems. The method systematically takes into account many-electron effects in the Mahan-Nozieres-De Dominicis (MND) theory, including core-level excitonic effects, the Fermi-edge singularity, shakeup excitations, and wave function overlap effects such as the orthogonality catastrophe, all within a universal framework using many-electron configurations. A heuristic search algorithm was introduced to search for the configurations that are important for defining the x-ray spectral line shape, instead of enumerating them in a brute-force way. The algorithm has proven to be efficient for calculating O K edges of transition metal oxides, which converge at the second excitation order (denoted as f((n)) with n = 2), i.e., the final-state configurations with two e-h pairs (with one hole being the core hole). However, it remains unknown how the determinant x-ray spectra converge for general cases and at which excitation order n should one stop the determinant calculation. Even with the heuristic algorithm, the number of many-electron configurations still grows exponentially with the excitation order n. In this work we prove two theorems that can indicate the order of magnitude of the contribution of the f((n)) configurations, so that one can estimate their contribution very quickly without actually calculating their amplitudes. The two theorems are based on singular-value decomposition (SVD) analysis, a method that is widely used to quantify entanglement between two quantum many-body systems. We examine the K edges of several metallic systems with the determinant formalism up to f((5)) to illustrate the usefulness of the theorems.