Abstract
The many-body perturbation theory methods based on the GW approximation and the Bethe-Salpeter equation (BSE) provide a first-principles route to modeling one- and two-particle excitations in a variety of bulk and molecular systems. This chapter reviews the current status of GW-BSE methods in the context of confined systems. We describe methods for basis set convergence, which allow sufficient numerical precision for accurate benchmarking of GW and BSE theory and study various theoretical approximations within GW. The differences in various flavors of GW and GW-BSE, including perturbative, self-consistent, and vertex-corrected implementations, are compared in the context of benchmark sets of sp-bonded aromatic molecules and Group IB and IIB transition metal atoms and ions with filled d shells.
References
Agostini F, Curchod B, Vuilleumier R, Tavernelli I, Gross EKU (2019) TDDFT and quantum-classical dynamics: a universal tool describing the dynamics of matter. In: Handbook of materials modeling. Volume 1 methods: theory and modeling, vol 1. Springer, Dordrecht
Anglade PM, Gonze X (2008) Preconditioning of self-consistent-field cycles in density-functional theory: the extrapolar method. Phys Rev B 78(4):045126. https://link.aps.org/doi/10.1103/PhysRevB.78.045126
Atalla V, Yoon M, Caruso F, Rinke P, Scheffler M (2013) Hybrid density functional theory meets quasiparticle calculations: a consistent electronic structure approach. Phys Rev B 88(16):165122. http://link.aps.org/doi/10.1103/PhysRevB.88.165122
Baumeier B, Andrienko D, Ma Y, Rohlfing M (2012) Excited states of Dicyanovinyl-substituted Oligothiophenes from many-body green’s functions theory. J Chem Theory Comput 8(3):997–1002. https://doi.org/10.1021/ct2008999
Baym G (1962) Self-consistent approximations in many-body systems. Phys Rev 127(4):1391–1401. http://link.aps.org/doi/10.1103/PhysRev.127.1391
Berger JA, Reining L, Sottile F (2010) Ab initio calculations of electronic excitations: collapsing spectral sums. Phys Rev B 82(4):041103. http://link.aps.org/doi/10.1103/PhysRevB.82.041103
Blase X, Attaccalite C (2011) Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach. Appl Phys Lett 99(17):171909. http://scitation.aip.org/content/aip/journal/apl/99/17/10.1063/1.3655352
Blase X, Attaccalite C, Olevano V (2011) First-principles GW calculations for fullerenes, porphyrins, phthalocyanine, and other molecules of interest for organic photovoltaic applications. Phys Rev B 83(11):115103. http://link.aps.org/doi/10.1103/PhysRevB.83.115103
Blase X, Boulanger P, Bruneval F, Fernandez-Serra M, Duchemin I (2016) GW and Bethe-Salpeter study of small water clusters. J Chem Phys 144(3):034109. http://scitation.aip.org/content/aip/journal/jcp/144/3/10.1063/1.4940139
Boulanger P, Jacquemin D, Duchemin I, Blase X (2014) Fast and accurate electronic excitations in Cyanines with the many-body Bethe–Salpeter approach. J Chem Theory Comput 10(3):1212–1218. https://doi.org/10.1021/ct401101u
Bruneval F (2012) Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies. J Chem Phys 136(19):194107. http://scitation.aip.org/content/aip/journal/jcp/136/19/10.1063/1.4718428
Bruneval F, Gonze X (2008) Accurate GW self-energies in a plane-wave basis using only a few empty states: towards large systems. Phys Rev B 78(8):085125. http://link.aps.org/doi/10.1103/PhysRevB.78.085125
Bruneval F, Marques MAL (2013) Benchmarking the starting points of the GW approximation for molecules. J Chem Theory Comput 9(1):324–329. https://doi.org/10.1021/ct300835h
Bruneval F, Sottile F, Olevano V, Del Sole R, Reining L (2005) Many-body perturbation theory using the density-functional concept: beyond the GW approximation. Phys Rev Lett 94(18):186402. http://link.aps.org/doi/10.1103/PhysRevLett.94.186402
Bruneval F, Hamed SM, Neaton JB (2015) A systematic benchmark of the ab initio Bethe-Salpeter equation approach for low-lying optical excitations of small organic molecules. J Chem Phys 142(24):244101. http://scitation.aip.org/content/aip/journal/jcp/142/24/10.1063/1.4922489
Bruneval F, Rangel T, Hamed SM, Shao M, Yang C, Neaton JB (2016) molgw 1: many-body perturbation theory software for atoms, molecules, and clusters. Comput Phys Commun 208:149–161. https://doi.org/10.1016/j.cpc.2016.06.019, http://www.sciencedirect.com/science/article/pii/S0010465516301990
Burrow PD, Michejda JA, Jordan KD (1987) Electron transmission study of the temporary negative ion states of selected Benzenoid and conjugated aromatic hydrocarbons. J Chem Phys 86(1):9–24. http://scitation.aip.org/content/aip/journal/jcp/86/1/10.1063/1.452598
Caruso F, Rinke P, Ren X, Rubio A, Scheffler M (2013) Self-consistent GW: all-electron implementation with localized basis functions. Phys Rev B 88(7):075105. http://link.aps.org/doi/10.1103/PhysRevB.88.075105
Casida ME (2009) Time-dependent density-functional theory for molecules and molecular solids. J Mol Struc Theochem 914(1–3):3–18. https://doi.org/10.1016/j.theochem.2009.08.018, http://www.sciencedirect.com/science/article/pii/S0166128009005363
Casida ME, Jamorski C, Casida KC, Salahub DR (1998) Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: characterization and correction of the time-dependent local density approximation ionization threshold. J Chem Phys 108(11):4439. https://doi.org/10.1063/1.475855, http://link.aip.org/link/JCPSA6/v108/i11/p4439/s1&Agg=doi
Dahlen NE, van Leeuwen R (2005) Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation. J Chem Phys 122(16):164102. http://scitation.aip.org/content/aip/journal/jcp/122/16/10.1063/1.1884965
Del Sole R, Reining L, Godby RW (1994) GWΓ approximation for electron self-energies in semiconductors and insulators. Phys Rev B 49(12):8024–8028. http://link.aps.org/doi/10.1103/PhysRevB.49.8024
Deslippe J, Samsonidze G, Strubbe DA, Jain M, Cohen ML, Louie SG (2012) BerkeleyGW: a massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Comput Phys Commun 183(6):1269–1289. http://www.sciencedirect.com/science/article/pii/S0010465511003912
Deslippe J, Samsonidze G, Jain M, Cohen ML, Louie SG (2013) Coulomb-hole summations and energies for GW calculations with limited number of empty orbitals: a modified static remainder approach. Phys Rev B 87(16):165124. http://link.aps.org/doi/10.1103/PhysRevB.87.165124
Faber C, Attaccalite C, Olevano V, Runge E, Blase X (2011) First-principles GW calculations for DNA and RNA nucleobases. Phys Rev B 83(11):115123. http://link.aps.org/doi/10.1103/PhysRevB.83.115123
Falden HH, Falster-Hansen KR, Bak KL, Rettrup S, Sauer SPA (2009) Benchmarking second order methods for the calculation of vertical electronic excitation energies: valence and Rydberg states in polycyclic aromatic hydrocarbons†. J Phys Chem A 113(43):11995–12012. https://doi.org/10.1021/jp9037123
Faleev SV, van Schilfgaarde M, Kotani T (2004) All-electron self-consistent GW approximation: application to Si, MnO, and NiO. Phys Rev Lett 93(12):126406. http://link.aps.org/doi/10.1103/PhysRevLett.93.126406
Fliegl H, Sundholm D (2014) Coupled-cluster calculations of the lowest 0–0 bands of the electronic excitation spectrum of naphthalene. Phys Chem Chem Phys 16(21):9859–9865. https://doi.org/10.1039/C3CP54421D, http://pubs.rsc.org/en/content/articlelanding/2014/cp/c3cp54421d
Gao W, Xia W, Gao X, Zhang P (2016) Speeding up GW calculations to meet the challenge of large scale Quasiparticle predictions. Sci Rep 6:36849. https://doi.org/10.1038/srep36849, http://www.nature.com/srep/2016/161111/srep36849/full/srep36849.html
Godby RW, Schlüter M, Sham LJ (1988) Self-energy operators and exchange-correlation potentials in semiconductors. Phys Rev B 37(17):10159–10175. http://link.aps.org/doi/10.1103/PhysRevB.37.10159
Grüneis A, Kresse G, Hinuma Y, Oba F (2014) Ionization potentials of solids: the importance of vertex corrections. Phys Rev Lett 112(9):096401. http://link.aps.org/doi/10.1103/PhysRevLett.112.096401
Gulans A (2014) Towards numerically accurate many-body perturbation theory: short-range correlation effects. J Chem Phys 141(16):164127. http://scitation.aip.org/content/aip/journal/jcp/141/16/10.1063/1.4900447
Hajgató B, Deleuze MS, Tozer DJ, De Proft F (2008) A benchmark theoretical study of the electron affinities of benzene and linear acenes. J Chem Phys 129(8):084308. http://scitation.aip.org/content/aip/journal/jcp/129/8/10.1063/1.2967182
Hedin L (1965) New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys Rev 139(3A):A796–A823. http://link.aps.org/doi/10.1103/PhysRev.139.A796
Hirose D, Noguchi Y, Sugino O (2015) All-electron GW+Bethe-Salpeter calculations on small molecules. Phys Rev B 91(20):205111. http://link.aps.org/doi/10.1103/PhysRevB.91.205111
Hung L, Baishya K, Öğüt S (2014) First-principles real-space study of electronic and optical excitations in rutile TiO2 nanocrystals. Phys Rev B 90(16):165424. http://link.aps.org/doi/10.1103/PhysRevB.90.165424
Hung L, da Jornada FH, Souto-Casares J, Chelikowsky JR, Louie SG, Öğüt S (2016) Excitation spectra of aromatic molecules within a real-space GW-BSE formalism: role of self-consistency and vertex corrections. Phys Rev B 94(8):085125. http://link.aps.org/doi/10.1103/PhysRevB.94.085125
Hung L, Bruneval F, Baishya K, Öğüt S (2017) Benchmarking the GW approximation and Bethe–Salpeter equation for groups IB and IIB atoms and monoxides. J Chem Theory Comput 13(5):2135–2146. https://doi.org/10.1021/acs.jctc.7b00123
Hybertsen MS, Louie SG (1986) Electron correlation in semiconductors and insulators: band gaps and quasiparticle energies. Phys Rev B 34(8):5390–5413. http://link.aps.org/doi/10.1103/PhysRevB.34.5390
Jacquemin D, Duchemin I, Blase X (2015) Benchmarking the Bethe–Salpeter formalism on a standard organic molecular set. J Chem Theory Comput 11(7):3290–3304. https://doi.org/10.1021/acs.jctc.5b00304
Jiang H, Blaha P (2016) GW with linearized augmented plane waves extended by high-energy local orbitals. Phys Rev B 93(11):115203. http://link.aps.org/doi/10.1103/PhysRevB.93.115203
Kang W, Hybertsen MS (2010) Enhanced static approximation to the electron self-energy operator for efficient calculation of quasiparticle energies. Phys Rev B 82(19):195108. http://link.aps.org/doi/10.1103/PhysRevB.82.195108
Kaplan F, Weigend F, Evers F, van Setten MJ (2015) Off-diagonal self-energy terms and partially self-consistency in GW calculations for single molecules: efficient implementation and quantitative effects on ionization potentials. J Chem Theory Comput 11(11):5152–5160. https://doi.org/10.1021/acs.jctc.5b00394
Kaplan F, Harding ME, Seiler C, Weigend F, Evers F, van Setten MJ (2016) Quasi-particle self-consistent GW for molecules. J Chem Theory Comput 12(6):2528–2541. https://doi.org/10.1021/acs.jctc.5b01238
Ke SH (2011) All-electron GW methods implemented in molecular orbital space: ionization energy and electron affinity of conjugated molecules. Phys Rev B 84(20):205415. http://link.aps.org/doi/10.1103/PhysRevB.84.205415
Klimeš J, Kaltak M, Kresse G (2014) Predictive GW calculations using plane waves and pseudopotentials. Phys Rev B 90(7):075125. http://link.aps.org/doi/10.1103/PhysRevB.90.075125
Knight JW, Wang X, Gallandi L, Dolgounitcheva O, Ren X, Ortiz JV, Rinke P, Körzdörfer T, Marom N (2016) Accurate ionization potentials and electron affinities of acceptor molecules III: a benchmark of GW methods. J Chem Theory Comput 12(2):615–626. https://doi.org/10.1021/acs.jctc.5b00871
Körbel S, Boulanger P, Duchemin I, Blase X, Marques MAL, Botti S (2014) Benchmark many-body GW and Bethe–Salpeter calculations for small transition metal molecules. J Chem Theory Comput 10(9):3934–3943. https://doi.org/10.1021/ct5003658
Koval P, Foerster D, Sánchez-Portal D (2014) Fully self-consistent GW and quasiparticle self-consistent $GW$ for molecules. Phys Rev B 89(15):155417. http://link.aps.org/doi/10.1103/PhysRevB.89.155417
Krause K, Harding ME, Klopper W (2015) Coupled-cluster reference values for the GW27 and GW100 test sets for the assessment of GW methods. Mol Phys 113(13–14):1952–1960. http://www.tandfonline.com/doi/full/10.1080/00268976.2015.1025113
Ku W, Eguiluz AG (2002) Band-gap problem in semiconductors revisited: effects of core states and many-body self-consistency. Phys Rev Lett 89(12):126401. http://link.aps.org/doi/10.1103/PhysRevLett.89.126401
Kutepov A, Savrasov SY, Kotliar G (2009) Ground-state properties of simple elements from GW calculations. Phys Rev B 80(4):041103. http://link.aps.org/doi/10.1103/PhysRevB.80.041103
Laurent AD, Jacquemin D (2013) TD-DFT benchmarks: a review. Int J Quantum Chem 113(17):2019–2039. http://onlinelibrary.wiley.com/doi/10.1002/qua.24438/abstract
Leang SS, Zahariev F, Gordon MS (2012) Benchmarking the performance of time-dependent density functional methods. J Chem Phys 136(10):104101. http://scitation.aip.org/content/aip/journal/jcp/136/10/10.1063/1.3689445
Louie SG, Rubio A (2005) Quasiparticle and optical properties of solids and nanostructures: the GW-BSE approach. In: Handbook of materials modeling. Springer, Dordrecht, pp 215–240. https://link.springer.com/chapter/10.1007/978-1-4020-3286-8_12
Ma Y, Rohlfing M, Molteni C (2010) Modeling the excited states of biological chromophores within many-body green’s function theory. J Chem Theory Comput 6(1):257–265. https://doi.org/10.1021/ct900528h
Maebashi H, Takada Y (2011) Analysis of exact vertex function for improving on the GWΓ scheme for first-principles calculation of electron self-energy. Phys Rev B 84(24):245134. http://link.aps.org/doi/10.1103/PhysRevB.84.245134
Marini A, Rubio A (2004) Electron linewidths of wide-gap insulators: excitonic effects in LiF. Phys Rev B 70(8):081103. http://link.aps.org/doi/10.1103/PhysRevB.70.081103
Marom N, Caruso F, Ren X, Hofmann OT, Körzdörfer T, Chelikowsky JR, Rubio A, Scheffler M, Rinke P (2012) Benchmark of GW methods for azabenzenes. Phys Rev B 86(24):245127. http://link.aps.org/doi/10.1103/PhysRevB.86.245127
Morris AJ, Stankovski M, Delaney KT, Rinke P, García-González P, Godby RW (2007) Vertex corrections in localized and extended systems. Phys Rev B 76(15):155106. http://link.aps.org/doi/10.1103/PhysRevB.76.155106
Onida G, Reining L, Rubio A (2002) Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev Mod Phys 74(2):601. http://link.aps.org/doi/10.1103/RevModPhys.74.601
Palmer MH (2008) The electronic states of 1,2,5-thiadiazole studied by VUV absorption spectroscopy and ab initio configuration interaction methods. Chem Phys 348(1–3):130–142. https://doi.org/10.1016/j.chemphys.2008.02.004, http://www.sciencedirect.com/science/article/pii/S0301010408001079
Pham TA, Nguyen HV, Rocca D, Galli G (2013) GW calculations using the spectral decomposition of the dielectric matrix: verification, validation, and comparison of methods. Phys Rev B 87:155148. http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.155148
Qian X, Umari P, Marzari N (2011) Photoelectron properties of DNA and RNA bases from many-body perturbation theory. Phys Rev B 84(7):075103. http://link.aps.org/doi/10.1103/PhysRevB.84.075103
Refaely-Abramson S, Baer R, Kronik L (2011) Fundamental and excitation gaps in molecules of relevance for organic photovoltaics from an optimally tuned range-separated hybrid functional. Phys Rev B 84(7):075144. http://link.aps.org/doi/10.1103/PhysRevB.84.075144
Ren X, Rinke P, Blum V, Wieferink J, Tkatchenko A, Sanfilippo A, Reuter K, Scheffler M (2012) Resolution-of-identity approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions. New J Phys 14(5):053020. https://doi.org/10.1088/1367-2630/14/5/053020, http://iopscience.iop.org/1367-2630/14/5/053020
Rocca D, Gebauer R, Saad Y, Baroni S (2008) Turbo charging time-dependent density-functional theory with Lanczos chains. J Chem Phys 128(15):154105. https://aip.scitation.org/doi/abs/10.1063/1.2899649
Rohlfing M, Louie SG (2000) Electron-hole excitations and optical spectra from first principles. Phys Rev B 62(8):4927–4944. http://link.aps.org/doi/10.1103/PhysRevB.62.4927
Romaniello P, Guyot S, Reining L (2009) The self-energy beyond GW: local and nonlocal vertex corrections. J Chem Phys 131(15):154111. http://scitation.aip.org/content/aip/journal/jcp/131/15/10.1063/1.3249965
Rostgaard C, Jacobsen KW, Thygesen KS (2010) Fully self-consistent GW calculations for molecules. Phys Rev B 81(8):085103. http://link.aps.org/doi/10.1103/PhysRevB.81.085103
Samsonidze G, Jain M, Deslippe J, Cohen ML, Louie SG (2011) Simple approximate physical orbitals for GW quasiparticle calculations. Phys Rev Lett 107(18):186404. http://link.aps.org/doi/10.1103/PhysRevLett.107.186404
Sharifzadeh S, Tamblyn I, Doak P, Darancet PT, Neaton JB (2012) Quantitative molecular orbital energies within a G0w0 approximation. Eur Phys J B 85(9):1–5. http://link.springer.com/article/10.1140/epjb/e2012-30206-0
Shirley EL (1996) Self-consistent GW and higher-order calculations of electron states in metals. Phys Rev B 54(11):7758–7764. http://link.aps.org/doi/10.1103/PhysRevB.54.7758
Stan A, Dahlen NE, van Leeuwen R (2006) Fully self-consistent GW calculations for atoms and molecules. Europhys Lett 76(2):298. https://doi.org/10.1209/epl/i2006-10266-6, http://iopscience.iop.org/0295-5075/76/2/298
Stan A, Dahlen NE, van Leeuwen R (2009) Levels of self-consistency in the GW approximation. J Chem Phys 130(11):114105. http://scitation.aip.org/content/aip/journal/jcp/130/11/10.1063/1.3089567
Stefanucci G, Pavlyukh Y, Uimonen AM, van Leeuwen R (2014) Diagrammatic expansion for positive spectral functions beyond GW: application to vertex corrections in the electron gas. Phys Rev B 90(11):115134. http://link.aps.org/doi/10.1103/PhysRevB.90.115134
Stenrup M (2012) Theoretical study of the radiationless deactivation mechanisms of photo-excited thiophene. Chem Phys 397:18–25. https://doi.org/10.1016/j.chemphys.2011.12.004, http://www.sciencedirect.com/science/article/pii/S0301010411005507
Strinati G (1988) Application of the green’s functions method to the study of the optical properties of semiconductors. Riv Nuovo Cimento 11(12):1–86
Tiago ML, Chelikowsky JR (2006) Optical excitations in organic molecules, clusters, and defects studied by first-principles Green’s function methods. Phys Rev B 73(20):205334. http://link.aps.org/doi/10.1103/PhysRevB.73.205334
Tozer DJ, Handy NC (1998) Improving virtual Kohn–Sham orbitals and eigenvalues: application to excitation energies and static polarizabilities. J Chem Phys 109(23):10180–10189. http://scitation.aip.org/content/aip/journal/jcp/109/23/10.1063/1.477711
Truhlar DG (1998) Basis-set extrapolation. Chem Phys Lett 294(1–3):45–48. https://doi.org/10.1016/S0009-2614(98)00866-5, http://www.sciencedirect.com/science/article/pii/S0009261498008665
Umari P, Stenuit G, Baroni S (2010) GW quasiparticle spectra from occupied states only. Phys Rev B 81(11):115104. http://link.aps.org/doi/10.1103/PhysRevB.81.115104
Ummels RTM, Bobbert PA, van Haeringen W (1998) First-order corrections to random-phase approximation GW calculations in silicon and diamond. Phys Rev B 57(19):11962–11973. http://link.aps.org/doi/10.1103/PhysRevB.57.11962
van Setten MJ, Caruso F, Sharifzadeh S, Ren X, Scheffler M, Liu F, Lischner J, Lin L, Deslippe JR, Louie SG, Yang C, Weigend F, Neaton JB, Evers F, Rinke P (2015) GW100: benchmarking G0w0 for molecular systems. J Chem Theory Comput 11(12):5665–5687. https://doi.org/10.1021/acs.jctc.5b00453
Wang LW (2015) Fully self-consistent solution of the Dyson equation using a plane-wave basis set. Phys Rev B 91(12):125135. http://link.aps.org/doi/10.1103/PhysRevB.91.125135
Acknowledgements
This work was supported by the US Department of Energy Grant No. DE-SC0017824 and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231, for computational resources.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this entry
Cite this entry
Hung, L., Öğüt, S. (2018). Modeling Excited States of Confined Systems. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling . Springer, Cham. https://doi.org/10.1007/978-3-319-42913-7_96-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-42913-7_96-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42913-7
Online ISBN: 978-3-319-42913-7
eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics