Local Potential Functional Embedding Theory of Molecular Systems: Localized Orbital-Based Embedding from an Exact Density-Functional Perspective

TL;DR

This work tackles the formal integration of density-functional theory with quantum embedding by formulating an exact density-functional embedding theory for lattice (and ab initio) Hamiltonians. It shows that fragment-dependent embedding potentials can be exactly related to the full-system KS Hxc potential, enabling a practical local potential functional embedding theory (LPFET) in which $v^{\rm Hxc}$ is the central variable and embedding clusters reproduce the KS density. The theory is instantiated into an IB (interacting bath) DMET framework for lattice DFT, with a local-density approximation based on the strong-correlation hypothesis. Numerical tests on a non-uniform six-site Hubbard ring, a Hubbard ladder, and an ab initio hydrogen chain demonstrate that LPFET improves density profiles in strongly correlated regimes and provides a path toward incorporating nonlocal exchange for better weakly correlated behavior.

Abstract

Localized orbital-based quantum embedding, as originally formulated in the context of density matrix embedding theory (DMET), is revisited from the perspective of lattice density functional theory (DFT). An in-principle exact (in the sense of full configuration interaction) formulation of the theory, where the occupations of the localized orbitals play the role of the density, is derived for any (model or ab initio) electronic Hamiltonian. From this general formalism we deduce an exact relation between the local Hartree-exchange-correlation (Hxc) potential of the full-size Kohn-Sham (KS) lattice-like system and the embedding chemical potential that is adjusted on each embedded fragment, individually, such that both KS and embedding cluster systems reproduce the exact same local density. When well-identified density-functional approximations (that find their justification in the strongly correlated regime) are applied, a practical self-consistent local potential functional embedding theory (LPFET), where the local Hxc potential becomes the basic variable, naturally emerges from the theory. LPFET differs from previous density embedding approaches by its fragment-dependent embedding chemical potential expression, which is a simple functional of the Hxc potential. Numerical calculations on prototypical systems show the ability of such an ansatz to improve substantially the description of density profiles (localized orbitals occupation numbers in this context) in strongly correlated systems.

Figures (12)

• Figure 1: Local density profiles obtained from self-consistent LPFET and DET calculations for weakly and moderately correlated half-filled 6-site Hubbard rings. The hopping parameter is set to $t=1$ and the external potential is fixed with $\max_i \left\{\vert v_i^{\rm ext}\vert\right\}=3$. Comparison is made with FCI. See text for further details.
• Figure 2: Same as Fig. \ref{['fig:weakly_strongly_correlated']} for stronger correlation regimes.
• Figure 3: Local density profiles obtained from self-consistent LPFET and DET calculations for weakly and moderately correlated half-filled 6-site Hubbard ladders. Comparison is made with FCI. See text for further details.
• Figure 4: Self-consistently converged Hxc potential profiles relative to site 0 ($\Delta v^{\text{Hxc}}_i=v^{\text{Hxc}}_i-v^{\text{Hxc}}_0$) obtained for the 6-site Hubbard ring with LPFET and DET methods in weakly and moderately correlated regimes. Comparison is made with the exact Hxc potential that has been determined from the FCI density profile.
• Figure 5: Same as Fig. \ref{['V_hxc_potentiel']} for stronger correlation regimes.