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Quantum embedding theories
Qiming Sun, Garnet Kin-Lic Chan
TL;DR
Quantum embedding addresses how to model a region of interest within a larger system while accounting for its environment at lower overall cost. The paper unifies density functional, Green’s function, and density matrix embedding through their shared theoretical structure, concluding that these methods will remain important for complex-system simulation while identifying frontier areas for development.
Problem
Existing reviews cover density functional, Green’s function, and density matrix embedding individually, but a unified presentation of their different formalisms is lacking.
Method
The paper introduces the basic equations and common intellectual structure of three rigorous embedding approaches based on the density, single-particle Green’s function, and density matrix.
Results
The account brings out the common intellectual structure of the three embedding approaches and concludes that quantum embedding methods will remain important for simulating complex systems.
Takeaways & Limitations
Quantum embedding provides a framework for focusing computation on a small region within a larger system while accounting for its environment.
Takeaways & Limitations
Frontier methodological needs include excited states and dynamics in density functional embedding, more efficient ab-initio technology in Green’s function and condensed-phase simulations, and sometimes classical embeddings.
Abstract
from arXiv · showhide
In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calculation on the active region of interest, and a low level calculation on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: it density functional embedding, Green's function embedding, and density matrix embedding, respectively use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used: functional equations for density functional embedding; diagrammatics for Green's function embedding; and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications, and possible future developments.
Introduction
Quantum embedding focuses computation on a fragment within a larger system while accounting for environmental effects. This account unifies three rigorous approaches by their quantum variables and emphasizes their shared structure, strengths, and weaknesses.
- Motivation: Embedding computes properties of a fragment while incorporating its environment without treating the full problem at full computational cost.The problem is partitioned into fragment A and environment B, though either may contain multiple partitions.
- Motivation: Theories differ by how environmental effects are communicated to the fragment and back to the environment.This communication is the defining distinction among embedding theories.
- Three embedding theories: The account covers density functional, Green’s function, and density matrix embedding as three related rigorous quantum embedding theories.They communicate information through the density, single-particle Green’s function, and single-particle density matrix, respectively.
- Three embedding theories: Embedding theories are identified by functional dependence on the relevant quantum variable, not by intermediary computational objects.A density matrix used in a calculation still corresponds to density functional embedding when it ultimately encodes a density functional.
- Paper scope: The paper presents all three formalisms together to highlight common intellectual strands and summarize their strengths and weaknesses.Existing reviews generally treat the techniques individually.
DFT embedding
DFT embedding derives an exact embedding potential that reproduces the subsystem density while connecting fragment and environment calculations through density-based variables. Practical approximations enable large environments and high-level wavefunction fragments, but accuracy is limited by kinetic, correlation, and interface errors.
- Formalism: DFT embedding partitions the full-system energy into fragment and remainder terms, deriving an embedding potential from stationarity with respect to the fragment density.The resulting potential is defined so fragment A satisfies an Euler equation in the environment field.
- Formalism: The exact embedding potential yields the exact subsystem density ρA.This establishes the formal density-level feedback between subsystem and environment.
- Formalism: The formalism commonly uses global and fragment densities, although practical DFT embedding often separates subsystem and environment densities with ρ = ρA + ρB.Separate densities can be advantageous when they are ensemble N-representable.
- Practical approximations: Approximating the kinetic energy with explicit density functionals simplifies the embedding potential, but accuracy is limited by the approximate kinetic energy.Numerical evaluation of the non-additive kinetic potential is a more recent alternative, though inversion can be difficult.
- Applications: DFT embedding has been most successful for weakly bound fragments, including van der Waals complexes, highly ionic crystals, and solvation.Frozen-environment variants can also treat very large environments such as protein frameworks.
- Wavefunction-in-DFT embedding: Wavefunction-in-DFT embedding treats the fragment with a high-level wavefunction method and the environment with DFT through the embedding potential.The environment effect is contained in vA, allowing existing quantum-chemistry programs to be adapted by adding this potential.
- Excited states: Excited-state embedding requires extending the ground-state formalism because using identical ground- and excited-state potentials is an approximation.State-specific approaches yield significant corrections when excited- and ground-state charge characters differ, while time-dependent DFT makes the embedding potential history-dependent.
- Limitations: Wavefunction-in-DFT embedding can suffer incomplete error cancellation or double counting, especially when the interface cuts a bond or omits important van der Waals interactions.Increasing the wavefunction-treated region can formally remedy these cases, but at increased cost.
Green’s function embedding
Green’s function embedding uses the single-particle Green’s function and its self-energy to connect fragment and environment descriptions. The hybridization is obtained explicitly from Green’s functions, while high- and low-level self-energy combinations enable applications from impurities to correlated materials but retain double-counting and non-local-correlation challenges.
- Formalism: The Green’s function encodes single-particle information, with its equal-time value giving the single-particle density matrix and its frequency representation describing spectral information.Unlike DFT, the energy can be computed explicitly from the exact Green’s function.
- Formalism: The Dyson equation relates non-interacting and interacting Green’s functions through a self-energy that accounts for interactions.Self-energy approximations select subsets of perturbation-theory diagrams and may be made self-consistent when G = GΣ.
- Embedding construction: Green’s function embedding adjusts the fragment Green’s function through a hybridization ΔA(ω), the Green’s-function analogue of the DFT embedding potential.The hybridization includes electron delocalization into the environment and environment Coulomb interactions.
- Embedding construction: G(ω) → Δ(ω) is obtained explicitly, so Green’s function embedding requires no iterative inversion technique.The isolated fragment Green’s function serves as the non-interacting reference for this inversion.
- Mean-field applications: At the mean-field level, savings arise by using different approximations for the fragment and full system or by foregoing environmental self-consistency.Applications include crystal impurities and molecular junctions with semi-infinite electrodes whose hybridization is assumed unchanged after molecule introduction.
- Correlated embedding: More sophisticated embedding combines a high-level fragment self-energy with a low-level full-system self-energy, analogous to wavefunction-in-DFT embedding.Self-consistency is obtained by solving the hybridization and composite-self-energy equations together.
- DMFT: DMFT is a widely used example in which strongly correlated fragment orbitals receive high-level self-energies within a lower-level full-system treatment.DFT+DMFT has been widely applied to correlated materials, especially for photoemission density-of-states calculations.
- Limitations: DFT+DMFT can suffer partial double counting, whereas Hartree-Fock-based combinations avoid this issue but omit correlations outside the fragments.The latter limitation prevents a quantitative description in the cited applications.
Density matrix embedding
Density matrix embedding treats the single-particle density matrix as its central quantum variable and represents an open fragment using a compact bath within a closed reference system. The method addresses the mismatch between fragment and closed-system density matrices, supports correlated treatment near fragments, and has applications to lattice and chemical systems.
- Framework: Density matrix embedding uses the single-particle density matrix as its central quantum variable and formally parallels density-functional and Green’s-function embedding.Its formulation starts from the density-matrix energy functional and stationarity conditions.
- Embedding equations: The fragment embedding operator encodes correlation effects beyond mean field, and stationarity yields the exact fragment density matrix.It is analogous to the exchange-correlation potential in DFT embedding and the self-energy in Green’s-function embedding.
- Framework: Open fragments generally have non-idempotent density matrices, so they cannot in general be matched by a closed non-interacting reference through an embedding operator.The formalism therefore considers interacting references or introduces bath degrees of freedom.
- Bath construction: Bath degrees of freedom embed the open fragment in a closed system, with the bath often substantially smaller than the environment.The same physical idea appears in impurity representations of Green’s-function embedding; link orbitals provide a simple QM/MM example.
- Bath construction: DMET provides a bath construction that is provably optimal at the mean-field level and is at most the same size as the fragment.This compresses most environmental degrees of freedom and avoids the much larger baths associated with Green’s-function embeddings.
- Bath construction: For a mean-field Slater determinant, the Schmidt decomposition identifies partially occupied bath orbitals from the environment block of the mean-field density matrix.The fragment-plus-bath reference system is non-interacting and has twice the fragment size.
- Scope and applications: DMET correlations are localized near each fragment, making the approach suitable for intensive quantities but requiring multiple fragments for extensive correlated treatments.Separate high-level fragment wavefunctions must then be assembled, with accuracy expected to be greatest near each fragment.
- Scope and applications: Applications include accurate large-fragment calculations for the 2D Hubbard model, non-trivial phases such as superconductivity, and dissociation curves of molecular chains and rings.The formalism also targets reduced-cost correlated calculations in solids, including small-band-gap systems and crystals in one, two, and three dimensions.
Conclusions
The paper unifies three quantum embedding approaches around their shared intellectual structure and surveys frontier methods, applications, and future directions.
- Quantum embedding provides a framework for simulating complex systems by focusing computation on a region within a larger system.
- The reviewed approaches use the single-particle density, Green’s function, and density matrix as their respective quantum variables.
- The account presents these three approaches together to emphasize their common intellectual structure.
- Frontier methodological areas include excited states and dynamics in density functional embedding and more efficient ab-initio technology in Green’s function and density matrix embedding.
- New application areas are emerging in biomolecular and condensed phase simulations, sometimes alongside classical embeddings such as QM/MM.
- Growing activity suggests that quantum embedding methods will remain important for simulating complex systems for many years.
Biographical information
The biographical information describes the academic and professional backgrounds of Qiming Sun and Garnet Kin-Lic Chan.
- Qiming Sun received his Ph.D. from Peking University and completed postdoctoral work at Princeton University.
- Sun joined the California Institute of Technology as a staff scientist and developed the PySCF quantum chemistry package.
- Garnet Kin-Lic Chan received his Ph.D. from the University of Cambridge and held postdoctoral positions at Christ’s College and the University of California, Berkeley.
- Chan held appointments at Cornell University and Princeton University before joining the California Institute of Technology.
- Chan is currently the Bren Professor of Chemistry.