From Full Dynamic to Pure Static: A Family of $GW$-Based Approximations

Published 9 Apr 2026 in physics.chem-ph, cond-mat.mtrl-sci, cond-mat.str-el, and nucl-th | (2604.08350v1)

Abstract: We introduce a systematic hierarchy of one-body Green's function methods derived from the $GW$ approximation, constructed by progressively reducing the dynamical content of the self-energy. Starting from the fully dynamical Dyson formulation, we generate a family of approximations that interpolates between the standard $GW$ approximation to purely static effective single-particle Hamiltonians. This framework enables a controlled investigation of the role of dynamical effects and particle-hole coupling in the description of ionization potentials. Within this unified formalism, the hole and particle branches can be selectively decoupled through downfolding strategies into reduced one-particle spaces. By benchmarking the different members of this hierarchy on molecular ionization energies, we assess their accuracy, numerical robustness, and algorithmic complexity. We demonstrate that consistently derived partially static schemes can yield reliable quasiparticle energies while significantly simplifying the underlying eigenvalue problem. We further introduce a novel static Hermitian self-energy obtained as the static limit of this hierarchy. Despite its conceptually distinct origin, it produces results remarkably close to those of qs$GW$, thereby providing an alternative static route toward partial self-consistency.

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Summary

The paper introduces a hierarchical framework that systematically quenches dynamical correlations in GW methods to connect diverse practical approaches.
It demonstrates that retaining minimal dynamical structure enables highly accurate prediction of ionization potentials while reducing computational demands.
The work resolves numerical instabilities in partially dynamical schemes using SRG-based regularization, advancing robust static and hybrid approximations.

Systematic Hierarchy of $GW$ -Based Approximations: From Full Dynamical to Purely Static Self-Energies

Motivation and Background

Charged excitation energies—ionization potentials (IPs) and electron affinities (EAs)—are central to predictive quantum chemistry and materials physics. They are uniquely challenging due to the interplay between dynamic correlation effects and the intrinsic complexity of the associated many-body eigenvalue problem. The $GW$ approximation for the one-body Green’s function, which incorporates dynamic screening of electron-electron interactions, has emerged as the standard tool for calculating single-particle excitation spectra of molecules and materials. However, a plethora of $GW$ variants with different treatments of the dynamical self-energy exist, including fully dynamical Dyson formulations, partially self-consistent approaches (ev $GW$ , qs $GW$ ), purely static approximations (COHSEX), diagonal quasiparticle equations, and non-Dyson ADC-style frameworks. Despite this diversity, there has been no unifying formal perspective connecting these schemes or systematically probing the consequences of reducing or removing the dynamical content of the self-energy.

This paper introduces a controlled hierarchy of $GW$ -based schemes by systematically quenching the dynamical character of the self-energy. This enables analysis of the effect of dynamical correlations and particle-hole mixing and clarifies the connections between the disparate practical $GW$ approaches.

Theoretical Framework: Supermatrix Formulation and Downfolding Strategies

The work adopts the supermatrix (extended Fock-space Green’s function) representation for $GW$ , which explicitly encompasses coupling between one-hole/one-particle ($1h$/$1p$) states and higher-order two-hole-one-particle (2h1p) and two-particle-one-hole (2p1h) excitations. This supermatrix leads to an effective Hamiltonian partitioned as:

$GW$ 0

where $GW$ 1 is the Fock matrix (split into occupied and virtual blocks), $GW$ 2 are coupling tensors, $GW$ 3 are block-diagonal configuration operators, and the off-diagonal supermatrix terms encode dynamical correlation via frequency-dependent self-energies.

By selective downfolding—integrating out certain classes of excitations—one can derive families of effective self-energies where the frequency dependence ('dynamical correlations') is retained, partially frozen, or fully quenched:

Fully dynamical (Dyson-like): Both 2h1p and 2p1h sectors are retained as frequency-dependent contributions.
Partially dynamical (half-and-half): Only one branch (hole or particle) is dynamical, the other is statically downfolded (evaluated at mean energy).
Fully static: Both branches are statically contracted, yielding a Hermitian static operator (pure static 'effective Hamiltonian').
Reduced/diagonal (non-Dyson, ADC-like): Restriction to the $GW$ 4 ( $GW$ 5) block, with or without static approximation, for efficient computation and explicit decoupling of IP and EA sectors.

This hierarchy also includes full diagonal approximations, as typically used in $GW$ 6, recovering popular practical methods as limiting cases.

Numerical Analysis: Accuracy, Robustness, and Regularization

A comprehensive molecular benchmark reveals that, for IPs of small molecules, the diagonal and one-branch-downfolded ("half-and-half") partially dynamical approximations maintain high fidelity with fully dynamical $GW$ 7. In contrast, the purely static limit, while numerically stable and conceptually simple, induces systematic shifts and loss of physical accuracy, especially for quantitative core-level energetics.

A critical finding is the identification of spurious large errors ("outliers") in some partially dynamical schemes, not attributable to the physical approximation but instead to numerical pathologies—divergences in the static evaluation of the self-energy denominator. These are resolved by introducing a similarity renormalization group (SRG)-based regularization that smoothly removes problematic terms without altering the robust schemes. After regularization, the partially dynamical (h{h}) approximation delivers mean absolute errors with respect to fully dynamical $GW$ 8 of less than 0.02 eV—a negligible error in routine applications.

Static Hermitian Self-Energy Construction and Relation to qs $GW$ 9

The authors introduce a novel static Hermitian self-energy operator, constructed by symmetrical static projection (evaluation at averaged energy), which differs conceptually from the commonly used qs $GW$ 0 (mode A) static symmetrization. Nonetheless, their numerical outcomes for IPs are extremely close (mean difference below 0.03 eV). This demonstrates a formal and practical equivalence between alternative static projections derived from the same hierarchy, providing new static routes towards partial self-consistency.

For deep-lying core states, all dynamical treatments converge, confirming the physical intuition that large energetic separation justifies the neglect of cross-sector dynamical mixing; only the pure static approximation again deviates substantially.

Implications: Decoupling Dynamical Correlation and Sector Coupling

By providing a unified framework, the work reveals that dynamical correlation and particle-hole sector coupling, often simultaneously changed in practical $GW$ 1 implementations, can instead be dialed independently. The minimal, but selective, retention of dynamical structure is sufficient for quantitative IPs in molecules—a key practical insight for scalable calculations.

Block-partitioned methods (ADC-like non-Dyson schemes) emerge as natural limiting cases in this hierarchy and, when consistently derived, can achieve accuracy comparable to full Dyson $GW$ 2 while dramatically reducing numerical complexity and eigenvalue targeting overheads.

The formalism also offers a valuable path for embedding and embedding-like approaches in which the dynamical content and configuration complexity must be tuned for tractable correlated calculations, especially in large or complex systems.

Conclusion

This work provides a systematic, physically motivated hierarchy of $GW$ 3-based Green’s function methods, interpolating between fully dynamical Dyson formulations and purely static effective single-particle Hamiltonians, and encompassing all major practical $GW$ 4-type approaches as limiting cases. The authors demonstrate that selective, consistent reduction of dynamical content enables reliable computation of molecular and core-level IPs, while block-partitioned and static schemes yield efficient practical methods. Critically, the analysis clarifies that observed failures in partially dynamical and non-Dyson schemes stem from removable numerical instabilities, rather than fundamental flaws. The hierarchy provides a map for balancing physical content, robustness, and computational tractability, opening avenues for next-generation scalable Green’s function methods and quantum embedding frameworks.