Fragment quantum embedding using the Householder transformation: a multi-state extension based on ensembles

TL;DR

This paper extends the Householder transformation-based quantum embedding method to multi-state systems, enabling accurate description of ground and excited states using ensemble density matrices in DMET.

Contribution

It introduces a multi-state extension of Householder-based embedding, allowing for exact ensemble embedding and a scalable number of bath orbitals proportional to fractional occupations.

Findings

Successfully applied to Hubbard lattice and hydrogen systems.

Demonstrated exact ensemble embedding with increased bath orbitals.

Connected ensemble embedding with traditional DMET bath construction.

Abstract

In recent works by Yalouz et al. (J. Chem. Phys. 157, 214112, 2022) and Sekaran et al. (Phys. Rev. B 104, 035121, 2021; Computation 10, 45, 2022), Density Matrix Embedding Theory (DMET) has been reformulated through the use of the Householder transformation as a novel tool to embed a fragment within extended systems. The transformation was applied to a reference non-interacting one-electron reduced density matrix to construct fragments' bath orbitals, which are crucial for subsequent ground state calculations. In the present work, we expand upon these previous developments and extend the utilization of the Householder transformation to the description of multiple electronic states, including ground and excited states. Based on an ensemble noninteracting density matrix, we demonstrate the feasibility of achieving exact fragment embedding through successive Householder transformations, resulting in a larger set of bath orbitals. We analytically prove that the number of additional bath orbitals scales directly with the number of fractionally occupied natural orbitals in the reference ensemble density matrix. A connection with the regular DMET bath construction is also made. Then, we illustrate the use of this ensemble embedding tool in single-shot DMET calculations to describe both ground and first excited states in a Hubbard lattice model and an ab initio hydrogen system. Lastly, we discuss avenues for enhancing ensemble embedding through self-consistency and explore potential future directions.