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The vertex function (Γ) within the Green’s function formalism encapsulates information about all higher-order electron–electron interaction beyond those mediated by density fluctuations. Herein, we present an efficient approach that embeds vertex corrections in the one-shot GW correlation self-energy for isolated and periodic systems. The vertex-corrected self-energy is constructed through the proposed separation–propagation–recombination procedure: the electronic Hilbert space is separated into an active space and its orthogonal complement denoted as the “rest;” the active component is propagated by a space-specific effective Hamiltonian different from the rest. The vertex corrections are introduced by a rescaled time-dependent nonlocal exchange interaction. The direct Γ correction to the self-energy is further updated by adjusting the rescaling factor in a self-consistent post-processing cycle. Our embedding method is tested mainly on donor–acceptor charge-transfer systems. The embedded vertex effects consistently and significantly correct the quasiparticle energies of the gap-edge states. The fundamental gap is generally improved by 1–3 eV upon the one-shot GW approximation. Furthermore, we provide an outlook for applications of (embedded) vertex corrections in calculations of extended solids.
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This content will become publicly available on June 28, 2026
Tensor hypercontraction for self-consistent vertex corrected GW with static and dynamic screening; applications to molecules and solids with superexchange
For molecules and solids, we developed efficient MPI-parallel algorithms for evaluating the second-order exchange (SOX) term with bare, statically screened, and dynamically screened interactions. We employ the resulting term in a fully self-consistent manner together with self-consistent GW (scGW), resulting in the following vertex-corrected scGW schemes: scGWSOX, scGWSOSEX, scGW2SOSEX, and scG3W2 theories. We show that for the vertex evaluation, the reduction of scaling by tensor hypercontraction has two limiting execution regimes. We used the resulting code to perform the largest (by the number of orbitals) fully self-consistent calculations with the SOX term. We demonstrate that our procedure allows for a reliable evaluation of even small energy differences. Utilizing a broken-symmetry approach, we explore the influence of the SOX term on the effective magnetic exchange couplings. We show that the treatment of SOX has a significant impact on the obtained values of the effective exchange constants, which we explain through a self-energy dependence on an effective dielectric constant. We confirm this explanation by analyzing natural orbitals and local changes in charge transfer, quantifying superexchange. Our analysis explains the structure of weak electron correlation responsible for the modulation of superexchange in both molecules and solids. Finally, for solids, we evaluate Néel temperatures utilizing the high-temperature expansion and compare the results obtained with experimental measurements. In addition, we prove a lack of Φ-derivability of the considered theories.
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- Award ID(s):
- 2154152
- PAR ID:
- 10633663
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- The Journal of Chemical Physics
- Volume:
- 162
- Issue:
- 24
- ISSN:
- 0021-9606
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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