Spectral Functions from Auxiliary-Field Quantum Monte Carlo without Analytic Continuation: The Extended Koopmans' Theorem Approach

TL;DR

This paper introduces a method to compute spectral functions directly from auxiliary-field quantum Monte Carlo without analytic continuation, improving accuracy for charge excitations in molecules and solids.

Contribution

It develops and benchmarks the extended Koopmans' theorem within AFQMC, enabling direct spectral function calculations and systematic error correction.

Findings

EKT1-AFQMC reproduces spectral features with <0.25 eV error

Method works well for small molecules and solids

Higher order EKT improves satellite region accuracy

Abstract

We explore the extended Koopmans' theorem (EKT) within the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method. The EKT allows for the direct calculation of electron addition and removal spectral functions using reduced density matrices of the $N$-particle system, and avoids the need for analytic continuation. The lowest level of EKT with AFQMC, called EKT1-AFQMC, is benchmarked using small molecules, 14-electron and 54-electron uniform electron gas supercells, and diamond at the $\Gamma$-point. Via comparison with numerically exact results (when possible) and coupled-cluster methods, we find that EKT1-AFQMC can reproduce the qualitative features of spectral functions for Koopmans-like charge excitations with errors in peak locations of less than 0.25 eV in a finite basis. We also note the numerical difficulties that arise in the EKT1-AFQMC eigenvalue problem, especially when back-propagated quantities are very noisy. We show how a systematic higher order EKT approach can correct errors in EKT1-based theories with respect to the satellite region of the spectral function. Our work will be of use for the study of low-energy charge excitations and spectral functions in correlated molecules and solids where AFQMC can be reliably performed.