1
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Patterson CH. Molecular Ionization Energies from GW and Hartree-Fock Theory: Polarizability, Screening, and Self-Energy Vertex Corrections. J Chem Theory Comput 2024; 20:7479-7493. [PMID: 39189378 PMCID: PMC11391582 DOI: 10.1021/acs.jctc.4c00795] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/28/2024]
Abstract
Accurate prediction of electron removal and addition energies is essential for reproducing neutral excitation spectra in molecules using Bethe-Salpeter equation methods. A Hartree-Fock starting point for GW/BSE calculations, combined with a random phase approximation (RPA) polarizability in the screened interaction, W, is well-known to overestimate neutral excitation energies. Using a Hartree-Fock starting point, we apply several different approximations for W to molecules in the Quest-3 database [Loos et al. J. Chem. Theory Comput. 2020, 16, 1711]. W is calculated using polarizabilities in RPA and time-dependent HF approximations. Inclusion of screened electron-hole attraction in the polarizability yields valence ionization energies in better agreement with experimental values and ADC(3) calculations than the more commonly applied RPA polarizability. Quasiparticle weights are also in better agreement with ADC(3) values when electron-hole attraction is included in W. Shake-up excitations for the 1π levels in benzene and azines are indicated only when electron-hole attraction is included. Ionization energies derived from HF eigenvalues via Koopmans theorem for molecules with nitrogen or oxygen lone pairs have the largest differences from experimental values in the molecules considered, leading to incorrect ordering of nonbonding and π bonding levels. Inclusion of electron-hole attraction in the polarizability results in correct ordering of ionization energies and marked improvement in agreement with experimental data. Vertex corrections to the self-energy further improve agreement with experimental ionization energies for these localized states.
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2
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Vacondio S, Varsano D, Ruini A, Ferretti A. Going Beyond the GW Approximation Using the Time-Dependent Hartree-Fock Vertex. J Chem Theory Comput 2024; 20:4718-4737. [PMID: 38772396 DOI: 10.1021/acs.jctc.4c00100] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/23/2024]
Abstract
The time-dependent Hartree-Fock (TDHF) vertex of many-body perturbation theory (MBPT) makes it possible to extend TDHF theory to charged excitations. Here we assess its performance by applying it to spherical atoms in their neutral electronic configuration. On a theoretical level, we recast the TDHF vertex as a reducible vertex, highlighting the emergence of a self-energy expansion purely in orders of the bare Coulomb interaction; then, on a numerical level, we present results for polarizabilities, ionization energies (IEs), and photoemission satellites. We confirm the superiority of THDF over simpler methods such as the random phase approximation for the prediction of atomic polarizabilities. We then find that the TDHF vertex reliably provides better IEs than GW and low-order self-energies do in the light-atom, few-electron regime; its performance degrades in heavier, many-electron atoms instead, where an expansion in orders of an unscreened Coulomb interaction becomes less justified. New relevant features are introduced in the satellite spectrum by the TDHF vertex, but the experimental spectra are not fully reproduced due to a missing account of nonlinear effects connected to hole relaxation. We also explore various truncations of the self-energy given by the TDHF vertex, but do not find them to be more convenient than low-order approximations such as GW and second Born (2B), suggesting that vertex corrections should be carried out consistently both in the self-energy and in the polarizability.
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Affiliation(s)
- Simone Vacondio
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Via G. Campi 213/a, 41125 Modena, Italy
- Centro S3, CNR-Istituto Nanoscienze, Via G. Campi 213/a, 41125 Modena, Italy
| | - Daniele Varsano
- Centro S3, CNR-Istituto Nanoscienze, Via G. Campi 213/a, 41125 Modena, Italy
| | - Alice Ruini
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Via G. Campi 213/a, 41125 Modena, Italy
- Centro S3, CNR-Istituto Nanoscienze, Via G. Campi 213/a, 41125 Modena, Italy
| | - Andrea Ferretti
- Centro S3, CNR-Istituto Nanoscienze, Via G. Campi 213/a, 41125 Modena, Italy
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3
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Wang X, Gao S, Luo Y, Liu X, Tom R, Zhao K, Chang V, Marom N. Computational Discovery of Intermolecular Singlet Fission Materials Using Many-Body Perturbation Theory. THE JOURNAL OF PHYSICAL CHEMISTRY. C, NANOMATERIALS AND INTERFACES 2024; 128:7841-7864. [PMID: 38774154 PMCID: PMC11103713 DOI: 10.1021/acs.jpcc.4c01340] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/29/2024] [Revised: 04/16/2024] [Accepted: 04/17/2024] [Indexed: 05/24/2024]
Abstract
Intermolecular singlet fission (SF) is the conversion of a photogenerated singlet exciton into two triplet excitons residing on different molecules. SF has the potential to enhance the conversion efficiency of solar cells by harvesting two charge carriers from one high-energy photon, whose surplus energy would otherwise be lost to heat. The development of commercial SF-augmented modules is hindered by the limited selection of molecular crystals that exhibit intermolecular SF in the solid state. Computational exploration may accelerate the discovery of new SF materials. The GW approximation and Bethe-Salpeter equation (GW+BSE) within the framework of many-body perturbation theory is the current state-of-the-art method for calculating the excited-state properties of molecular crystals with periodic boundary conditions. In this Review, we discuss the usage of GW+BSE to assess candidate SF materials as well as its combination with low-cost physical or machine learned models in materials discovery workflows. We demonstrate three successful strategies for the discovery of new SF materials: (i) functionalization of known materials to tune their properties, (ii) finding potential polymorphs with improved crystal packing, and (iii) exploring new classes of materials. In addition, three new candidate SF materials are proposed here, which have not been published previously.
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Affiliation(s)
- Xiaopeng Wang
- School
of Foundational Education, University of
Health and Rehabilitation Sciences, Qingdao 266113, China
- Qingdao
Institute for Theoretical and Computational Sciences, Institute of
Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, P. R. China
| | - Siyu Gao
- Department
of Materials Science and Engineering, Carnegie
Mellon University, Pittsburgh, Pennsylvania 15213, United States
| | - Yiqun Luo
- Department
of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States
| | - Xingyu Liu
- Department
of Materials Science and Engineering, Carnegie
Mellon University, Pittsburgh, Pennsylvania 15213, United States
| | - Rithwik Tom
- Department
of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States
| | - Kaiji Zhao
- Department
of Materials Science and Engineering, Carnegie
Mellon University, Pittsburgh, Pennsylvania 15213, United States
| | - Vincent Chang
- Department
of Materials Science and Engineering, Carnegie
Mellon University, Pittsburgh, Pennsylvania 15213, United States
| | - Noa Marom
- Department
of Materials Science and Engineering, Carnegie
Mellon University, Pittsburgh, Pennsylvania 15213, United States
- Department
of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States
- Department
of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States
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4
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Bruneval F, Förster A. Fully Dynamic G3 W2 Self-Energy for Finite Systems: Formulas and Benchmark. J Chem Theory Comput 2024; 20:3218-3230. [PMID: 38603811 DOI: 10.1021/acs.jctc.4c00090] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/13/2024]
Abstract
Over the years, Hedin's GW self-energy has been proven to be a rather accurate and simple approximation to evaluate electronic quasiparticle energies in solids and in molecules. Attempts to improve over the simple GW approximation, the so-called vertex corrections, have been constantly proposed in the literature. Here, we derive, analyze, and benchmark the complete second-order term in the screened Coulomb interaction W for finite systems. This self-energy named G3W2 contains all the possible time orderings that combine 3 Green's functions G and 2 dynamic W. We present the analytic formula and its imaginary frequency counterpart, with the latter allowing us to treat larger molecules. The accuracy of the G3W2 self-energy is evaluated on well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. Its link with the simpler static approximation, named SOSEX for static screened second-order exchange, is analyzed, which leads us to propose a more consistent approximation named 2SOSEX. In the end, we find that neither the G3W2 self-energy nor any of the investigated approximations to it improve over one-shot G0W0 with a good starting point. Only quasi-particle self-consistent GW HOMO energies are slightly improved by addition of the G3W2 self-energy correction. We show that this is due to the self-consistent update of the screened Coulomb interaction, leading to an overall sign change of the vertex correction to the frontier quasiparticle energies.
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Affiliation(s)
- Fabien Bruneval
- Université Paris-Saclay, CEA, Service de recherche en Corrosion et Comportement des Matériaux, SRMP, 91191 Gif-sur-Yvette, France
| | - Arno Förster
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1108, 1081 HZ Amsterdam, The Netherlands
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5
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Wen M, Abraham V, Harsha G, Shee A, Whaley KB, Zgid D. Comparing Self-Consistent GW and Vertex-Corrected G0W0 ( G0W0Γ) Accuracy for Molecular Ionization Potentials. J Chem Theory Comput 2024; 20:3109-3120. [PMID: 38573104 DOI: 10.1021/acs.jctc.3c01279] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/05/2024]
Abstract
We test the performance of self-consistent GW and several representative implementations of vertex-corrected G0W0 (G0W0Γ). These approaches are tested on benchmark data sets covering full valence spectra (first ionization potentials and some inner valence shell excitations). For small molecules, when comparing against state-of-the-art wave function techniques, our results show that full self-consistency in the GW scheme either systematically outperforms vertex-corrected G0W0 or gives results of at least comparative quality. Moreover, G0W0Γ results in additional computational cost when compared to G0W0 or self-consistent GW. The dependency of G0W0Γ on the starting mean-field solution is frequently more dominant than the magnitude of the vertex correction itself. Consequently, for molecular systems, self-consistent GW performed on the imaginary axis (and then followed by modern analytical continuation techniques) offers a more reliable approach to make predictions of ionization potentials.
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Affiliation(s)
- Ming Wen
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Vibin Abraham
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Gaurav Harsha
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Avijit Shee
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - K Birgitta Whaley
- Department of Chemistry, University of California, Berkeley, California 94720-1460, United States
| | - Dominika Zgid
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, United States
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6
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Orlando R, Romaniello P, Loos PF. The three channels of many-body perturbation theory: GW, particle-particle, and electron-hole T-matrix self-energies. J Chem Phys 2023; 159:184113. [PMID: 37962450 DOI: 10.1063/5.0176898] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2023] [Accepted: 10/23/2023] [Indexed: 11/15/2023] Open
Abstract
We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known GW self-energy, as well as the particle-particle and electron-hole T-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (B2 and C2) is also discussed.
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Affiliation(s)
- Roberto Orlando
- Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, Toulouse, France
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France
- European Theoretical Spectroscopy Facility (ETSF)
| | - Pina Romaniello
- Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France
- European Theoretical Spectroscopy Facility (ETSF)
| | - Pierre-François Loos
- Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, Toulouse, France
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7
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Förster A, van Lenthe E, Spadetto E, Visscher L. Two-Component GW Calculations: Cubic Scaling Implementation and Comparison of Vertex-Corrected and Partially Self-Consistent GW Variants. J Chem Theory Comput 2023; 19:5958-5976. [PMID: 37594901 PMCID: PMC10501001 DOI: 10.1021/acs.jctc.3c00512] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2023] [Indexed: 08/20/2023]
Abstract
We report an all-electron, atomic orbital (AO)-based, two-component (2C) implementation of the GW approximation (GWA) for closed-shell molecules. Our algorithm is based on the space-time formulation of the GWA and uses analytical continuation (AC) of the self-energy, and pair-atomic density fitting (PADF) to switch between AO and auxiliary basis. By calculating the dynamical contribution to the GW self-energy at a quasi-one-component level, our 2C-GW algorithm is only about a factor of 2-3 slower than in the scalar relativistic case. Additionally, we present a 2C implementation of the simplest vertex correction to the self-energy, the statically screened G3W2 correction. Comparison of first ionization potentials (IPs) of a set of 67 molecules with heavy elements (a subset of the SOC81 set) calculated with our implementation against results from the WEST code reveals mean absolute deviations (MAD) of around 70 meV for G0W0@PBE and G0W0@PBE0. We check the accuracy of our AC treatment by comparison to full-frequency GW calculations, which shows that in the absence of multisolution cases, the errors due to AC are only minor. This implies that the main sources of the observed deviations between both implementations are the different single-particle bases and the pseudopotential approximation in the WEST code. Finally, we assess the performance of some (partially self-consistent) variants of the GWA for the calculation of first IPs by comparison to vertical experimental reference values. G0W0@PBE0 (25% exact exchange) and G0W0@BHLYP (50% exact exchange) perform best with mean absolute deviations (MAD) of about 200 meV. Explicit treatment of spin-orbit effects at the 2C level is crucial for systematic agreement with experiment. On the other hand, eigenvalue-only self-consistent GW (evGW) and quasi-particle self-consistent GW (qsGW) significantly overestimate the IPs. Perturbative G3W2 corrections increase the IPs and therefore improve the agreement with experiment in cases where G0W0 alone underestimates the IPs. With a MAD of only 140 meV, 2C-G0W0@PBE0 + G3W2 is in best agreement with the experimental reference values.
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Affiliation(s)
- Arno Förster
- Theoretical
Chemistry, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
| | - Erik van Lenthe
- Software
for Chemistry and Materials NV, 1081 HV Amsterdam, The Netherlands
| | - Edoardo Spadetto
- Software
for Chemistry and Materials NV, 1081 HV Amsterdam, The Netherlands
| | - Lucas Visscher
- Theoretical
Chemistry, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
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8
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Weng G, Mallarapu R, Vlček V. Embedding vertex corrections in GW self-energy: Theory, implementation, and outlook. J Chem Phys 2023; 158:144105. [PMID: 37061461 DOI: 10.1063/5.0139117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/17/2023] Open
Abstract
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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Affiliation(s)
- Guorong Weng
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, USA
| | - Rushil Mallarapu
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, USA
| | - Vojtěch Vlček
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, USA
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9
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Förster A. Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies. J Chem Theory Comput 2022; 18:5948-5965. [PMID: 36150190 PMCID: PMC9558381 DOI: 10.1021/acs.jctc.2c00366] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
![]()
With increasing interelectronic distance, the screening
of the
electron–electron interaction by the presence of other electrons
becomes the dominant source of electron correlation. This effect is
described by the random phase approximation (RPA) which is therefore
a promising method for the calculation of weak interactions. The success
of the RPA relies on the cancellation of errors, which can be traced
back to the violation of the crossing symmetry of the 4-point vertex,
leading to strongly overestimated total correlation energies. By the
addition of second-order screened exchange (SOSEX) to the correlation
energy, this issue is substantially reduced. In the adiabatic connection
(AC) SOSEX formalism, one of the two electron–electron interaction
lines in the second-order exchange term is dynamically screened (SOSEX(W, vc)). A
related SOSEX expression in which both electron–electron interaction
lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, vc), the
evaluation of this correlation energy expression does not require
an expensive numerical frequency integration and is therefore advantageous
from a computational perspective. We compare the accuracy of the statically
screened variant to RPA and RPA+SOSEX(W, vc) for a wide range of chemical
reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, vc) for
charged excitations and noncovalent interactions where they lead to
major improvements over RPA.
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Affiliation(s)
- Arno Förster
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV, Amsterdam, The Netherlands
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10
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Vacondio S, Varsano D, Ruini A, Ferretti A. Numerically Precise Benchmark of Many-Body Self-Energies on Spherical Atoms. J Chem Theory Comput 2022; 18:3703-3717. [PMID: 35561415 PMCID: PMC9202310 DOI: 10.1021/acs.jctc.2c00048] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
![]()
We investigate the
performance of beyond-GW approaches in many-body
perturbation theory by addressing atoms described within the spherical
approximation via a dedicated numerical treatment based on B-splines
and spherical harmonics. We consider the GW, second Born (2B), and
GW + second order screened exchange (GW+SOSEX) self-energies and use
them to obtain ionization potentials from the quasi-particle equation
(QPE) solved perturbatively on top of independent-particle calculations.
We also solve the linearized Sham–Schlüter equation
(LSSE) and compare the resulting xc potentials against exact data.
We find that the LSSE provides consistent starting points for the
QPE but does not present any practical advantage in the present context.
Still, the features of the xc potentials obtained with it shed light
on possible strategies for the inclusion of beyond-GW diagrams in
the many-body self-energy. Our findings show that solving the QPE
with the GW+SOSEX self-energy on top of a PBE or PBE0 solution is
a viable scheme to go beyond GW in finite systems, even in the atomic
limit. However, GW shows a comparable performance if one agrees to
use a hybrid starting point. We also obtain promising results with
the 2B self-energy on top of Hartree–Fock, suggesting that
the full time-dependent Hartree–Fock vertex may be another
viable beyond-GW scheme for finite systems.
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Affiliation(s)
- S Vacondio
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via G. Campi 213/a, Modena 41121, Italy.,Centro S3, CNR-Istituto Nanoscienze, 41125 Modena, Italy
| | - D Varsano
- Centro S3, CNR-Istituto Nanoscienze, 41125 Modena, Italy
| | - A Ruini
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via G. Campi 213/a, Modena 41121, Italy.,Centro S3, CNR-Istituto Nanoscienze, 41125 Modena, Italy
| | - A Ferretti
- Centro S3, CNR-Istituto Nanoscienze, 41125 Modena, Italy
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11
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Bruneval F, Dattani N, van Setten MJ. The GW Miracle in Many-Body Perturbation Theory for the Ionization Potential of Molecules. Front Chem 2022; 9:749779. [PMID: 35004607 PMCID: PMC8733722 DOI: 10.3389/fchem.2021.749779] [Citation(s) in RCA: 27] [Impact Index Per Article: 13.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2021] [Accepted: 09/14/2021] [Indexed: 11/30/2022] Open
Abstract
We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chemistry methods. First of all, we revisit the reference CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 molecules, we calculate the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and, GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approximations for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (WTDHF), and with a GW density-matrix (γGW). Only the γGW result has a positive impact. Finally using an improved hybrid functional for the non-interacting Green’s function, considering it as a cheap way to approximate self-consistency, the accuracy of the simplest GW approximation improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.
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Affiliation(s)
- Fabien Bruneval
- CEA, Service de Recherches de Métallurgie Physique, Direction des Energies, Université Paris-Saclay, Paris, France
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