2,925
Views
2
CrossRef citations to date
0
Altmetric
Reviews

Ab initio modeling of excitons: from perfect crystals to biomaterials

ORCID Icon & ORCID Icon
Article: 1912638 | Received 05 Nov 2020, Accepted 29 Mar 2021, Published online: 02 May 2021

References

  • Lanzani G. The photophysics behind photovoltaics and photonics. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA; 2012.
  • Förster T. Zwischenmolekulare Energiewanderung und Fluoreszenz. Ann Phys. 1948 Jan;437:55–38.
  • Dexter DL. A theory of sensitized luminescence in solids. J Chem Phys. 1953 May;21:836–850.
  • Valeur B, Berberan-Santos MN. Molecular fluorescence. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA; 2012.
  • Kancherla R, Muralirajan K, Sagadevan A, et al. Visible light-induced excited-state transition-metal catalysis. Science. 2019;1:510.
  • Halls JJ, Walsh CA, Greenham NC, et al. Efficient photodiodes from interpenetrating polymer networks. Nature. 1995;376:498–500.
  • Helms V. Principles of computational cell biology: from protein complexes to cellular networks. Weinheim: Wiley; 2018.
  • Tang CW, Vanslyke SA. Organic electroluminescent diodes. Appl Phys Lett. 1987;51:913–915.
  • Baumeier B, May F, Lennartz C, et al. Challenges for in silico design of organic semiconductors. J Mater Chem. 2012 Jun;22:10971–10976.
  • Adachi C. Third-generation organic electroluminescence materials. Jpn J Appl Phys. 2014 May;53:060101.
  • Streetman B. Solid state electronic devices. 5th ed. Upper Saddle River N.J.: Prentice Hall; 2000.
  • Wannier GH. The structure of electronic excitation levels in insulating crystals. Phys Rev. 1937 Aug;52:191–197.
  • Frenkel J. On the transformation of light into heat in solids. Science. 1931 Jan;37:17–44.
  • Wright J. Molecular crystals. Cambridge: Cambridge University Press; 1995.
  • Hodes G. Perovskite-based solar cells. Science. 2013 Oct;342:317–318.
  • Howard IA, Meister M, Baumeier B, et al. Two channels of charge generation in perylene monoimide solid-state dye-sensitized solar cells. Adv Energy Mater. 2014 Jan;4:1300640.
  • Baumeier B, Rohlfing M, Andrienko D. Electronic excitations in push-pull oligomers and their complexes with fullerene from many-body Green’s functions theory with polarizable embedding. J Chem Theory Comput. 2014 Aug;10:3104–3110.
  • Bagheri B, Baumeier B, Karttunen M. Getting excited: challenges in quantum-classical studies of excitons in polymeric systems. Phys Chem Chem Phys. 2016 Nov;18:30297–30304.
  • Bagheri B, Karttunen M, Baumeier B. Solvent effects on optical excitations of poly para phenylene ethynylene studied by QM/MM simulations based on many-body Green’s functions theory. Eur Phys J Spec Top. 2016 Dec;225:1743–1756.
  • Forrest SR. Excitons and the lifetime of organic semiconductor devices. Philos Trans R Soc A Math Phys Eng Sci. 2015 Jun;373:2044.
  • Born M, Oppenheimer R. Zur Quantentheorie der Molekeln. Ann Phys. 1927 Jan;389:457–484.
  • Martin RM, Press CU. Electronic structure: basic theory and practical methods. Cambridge: Cambridge University Press; 2004.
  • Cramer C. Essentials of computational chemistry: theories and models. Weinheim: Wiley; 2013.
  • Pal, S., Prasad, M.D. & Mukherjee, D. On certain correspondences among various coupled-cluster theories for closed-shell systems. Pramana - J. Phys. 18, 261–270 (1982).
  • Van Voorhis T, Head-Gordon M. Benchmark variational coupled cluster doubles results. J Chem Phys. 2000 Nov;113:8873–8879.
  • Cooper B, Knowles PJ. Benchmark studies of variational, unitary and extended coupled cluster methods. J Chem Phys. 2010 Dec;133:234102.
  • Evangelista FA. Alternative single-reference coupled cluster approaches for multireference problems: the simpler, the better. J Chem Phys. 2011 Jun;134:224102.
  • Jeziorski B, Monkhorst HJ. Coupled-cluster method for multideterminantal reference states. Phys Rev A. 1981 Oct;24:1668–1681.
  • Lindgren I, Mukherjee D. On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces. Phys Rep. 1987 Jul;151:93–127.
  • Sneskov K, Christiansen O. Excited state coupled cluster methods. Wiley Interdiscip Rev Comput Mol Sci. 2012;2:566–584.
  • Wang X, Berkelbach TC. Excitons in solids from periodic equation-of-motion coupled-cluster theory. J Chem Theory Comput. 2020 May;16:3095–3103.
  • Gruber T, Liao K, Tsatsoulis T, et al. Applying the coupled-cluster ansatz to solids and surfaces in the thermodynamic limit. Phys Rev X. 2018;8:021043.
  • Hummel F. Finite temperature coupled cluster theories for extended systems. J Chem Theory Comput. 2018;14:6505–6514.
  • Lange MF, Berkelbach TC. Active space approaches combining coupled-cluster and perturbation theory for ground states and excited states. Mol Phys. 2020 oct;118:e1808726.
  • Gao Y, Sun Q, Yu JM, et al. Electronic structure of bulk manganese oxide and nickel oxide from coupled cluster theory. Phys Rev B. 2020 Apr;101:165138.
  • Zhao L, Neuscamman E. Variational excitations in real solids: optical gaps and insights into many-body perturbation theory. Phys Rev Lett. 2019 Jul;123:036402.
  • Williamson A, Hood RQ, Needs R, et al. Diffusion quantum Monte Carlo calculations of the excited states of silicon. Phys Rev Lett. 1998 May;57:12140–12144.
  • Hunt RJ, Szyniszewski M, Prayogo GI, et al. Quantum Monte Carlo calculations of energy gaps from first principles. Phys Rev B. 2018 Aug;98:075122.
  • Ma F, Zhang S, Krakauer H. Excited state calculations in solids by auxiliary-field quantum Monte Carlo. New J Phys. 2013 Sep;15:093017.
  • Onida G, Reining L, Rubio A. Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev Mod Phys. 2002 June;74:601–659.
  • Rohlfing M, Louie SG. Electron-hole excitations in semiconductors and insulators. Phys Rev Lett. 1998 Sep;81:2312–2315.
  • Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys Rev. 1964;136:B864. [ Nov].
  • Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys Rev. 1965 Nov;140:A1133.
  • Burke K. Perspective on density functional theory. J Chem Phys. 2012 Apr;136:150901.
  • Tsuneda T. Exchange-Correlation Functionals. In:density functional theory in quantum chemistry. Japan: Springer; 2014. p. 101–124.
  • Perdew JP. Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys Rev B. 1986 Jun;33:8822–8824.
  • Perdew JP. Erratum: density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys Rev B. 1986 Nov;34:7406.
  • Lee C, Yang W, Parr RG. Development of the colle-salvetti correlation-energy formula into a functional of the electron density. Phys Rev B. 1988 Jan;37:785–789.
  • Becke AD. Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys. 1993 Apr;98:5648–5652.
  • Perdew JP. Density functional theory and the band gap problem. Int J Quantum Chem. 2009 Jun;28:497–523.
  • Engel E, Dreizler RM. Density functional theory. Berlin, Heidelberg: Springer Berlin Heidelberg; 2011. https://www.springer.com/gp/book/9783642140891.
  • Runge E, Gross EK. Density-functional theory for time-dependent systems. Phys Rev Lett. 1984 Mar;52:997–1000.
  • Gross EK, Kohn W. Local density-functional theory of frequency-dependent linear response. Phys Rev Lett. 1985 Dec;55:2850–2852.
  • Casida ME. Time-dependent density functional response theory for molecules. Singapore: World Scientific; 1995.
  • Foerster D. Fast computation of the Kohn-Sham susceptibility of large systems. Phys Rev Lett. 2005 Aug;72:073106.
  • Hirata S, Head-Gordon M. Time-dependent density functional theory within the Tamm-Dancoff approximation. Chem Phys Lett. 1999 Dec;314:291–299.
  • Furche F, Ahlrichs R. Adiabatic time-dependent density functional methods for excited state properties. J Chem Phys. 2002 Sep;117:7433–7447.
  • Menger MF, Caprasecca S, Mennucci B. Excited-state gradients in polarizable QM/MM models: an induced dipole formulation. J Chem Theory Comput. 2017 Aug;13:3778–3786.
  • Paier J, Marsman M, Kresse G. Dielectric properties and excitons for extended systems from hybrid functionals. Phys Rev Lett. 2008 Sep;78:121201.
  • Sharma S, Dewhurst JK, Sanna A, et al. Bootstrap approximation for the exchange-correlation kernel of time-dependent density-functional theory. Phys Rev Lett. 2011 Oct;107:186401.
  • Rigamonti S, Botti S, Veniard V, et al. Estimating excitonic effects in the absorption spectra of solids: problems and insight from a guided iteration scheme. Phys Rev Lett. 2015 Apr;114:146402.
  • Reining L, Olevano V, Rubio A, Reining L, Olevano V, Rubio A, et al. Excitonic effects in solids described by time-dependent density-functional theory. Phys Rev Lett. 2002 Jan;88:066404.
  • Kümmel S. Charge-transfer excitations: a challenge for time-dependent density functional theory that has been met. Adv Energy Mater. 2017 Aug;7:1700440.
  • Casanova-Páez M, Goerigk L. Assessing the Tamm-Dancoff approximation, singlet-singlet, and singlet-triplet excitations with the latest long-range corrected double-hybrid density functionals. J Chem Phys. 2020 Aug;153:64106.
  • Hedin L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys Rev. 1965 Aug;139:A796.
  • Strinati G. Application of the Green’s functions method to the study of the optical properties of semiconductors. Science. 1988 Dec;11:1–86.
  • Hybertsen MS, Louie SG. Electron correlation in semiconductors and insulators: band gaps and quasiparticle energies. Phys Rev B. 1986 Oct;34:5390–5413.
  • Aulbur WG, Städele M, Görling A. Exact-exchange-based quasiparticle calculations. Phys Rev Lett. 2000 Sep;62:7121–7132.
  • Rohlfing M, Louie SG. Electron-hole excitations and optical spectra from first principles. Phys Rev Lett. 2000 Aug;62:4927–4944.
  • Van Schilfgaarde M, Kotani T, Faleev S. Quasiparticle self-consistent gw theory. Phys Rev Lett. 2006 Jun;96:226402.
  • Stan A, Dahlen NE, Van Leeuwen R. Levels of self-consistency in the GW approximation. J Chem Phys. 2009 Mar;130:114105.
  • Golze D, Dvorak M, Rinke P. The GW compendium: a practical guide to theoretical photoemission spectroscopy. Front Chem. 2019 Jul;7:377.
  • Von Der Linden W, Horsch P. Precise quasiparticle energies and Hartree-Fock bands of semiconductors and insulators. Phys Rev B. 1988 May;37:8351–8362.
  • Godby RW, Needs RJ. Metal-insulator transition in Kohn-Sham theory and quasiparticle theory. Phys Rev Lett. 1989 Mar;62:1169–1172.
  • Engel GE, Farid B. Generalized plasmon-pole model and plasmon band structures of crystals. Phys Rev B. 1993 Jun;47:15931–15934.
  • Larson P, Dvorak M, Wu Z. Role of the plasmon-pole model in the GW approximation. Phys Rev Lett. 2013 Sep;88:125205.
  • Stankovski M, Antonius G, Waroquiers D, et al. G0W0 band gap of ZnO: effects of plasmon-pole models. Phys Rev Lett. 2011 Dec;84:241201.
  • Godby RW, Schlüter M, Sham LJ. Self-energy operators and exchange-correlation potentials in semiconductors. Phys Rev B. 1988 Jun;37:10159–10175.
  • Bloechl PE, Lebègue S, Alouani M, et al. Implementation of an all-electron GW approximation based on the projector augmented wave method without plasmon pole approximation: application to Si, SiC, AlAs, InAs, NaH, and KH. Phys Rev Lett. 2003 Apr;67:155208.
  • Kotani T, Van Schilfgaarde M, Faleev SV. Quasiparticle self-consistent GW method: a basis for the independent-particle approximation. Phys Rev Lett. 2007 Oct;76:165106.
  • Gonze X, Amadon B, Anglade PM, et al. ABINIT: first-principles approach to material and nanosystem properties. Comput Phys Commun. 2009 Dec;180:2582–2615.
  • Blase X, Attaccalite C, Olevano V. First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications. Phys Rev Lett. 2011 Mar;83:115103.
  • Golze D, Wilhelm J, Van Setten MJ, et al. Core-level binding energies from GW: an efficient full-frequency approach within a localized basis. J Chem Theory Comput. 2018 Sep;14:4856–4869.
  • Rieger MM, Steinbeck L, White ID, et al. GW space-time method for the self-energy of large systems. Comput Phys Commun. 1999 Mar;117:211–228.
  • Friedrich C, Blügel S, Schindlmayr A. Efficient implementation of the GW approximation within the all-electron FLAPW method. Phys Rev Lett. 2010 Mar;81:125102.
  • Van Setten MJ, Caruso F, Sharifzadeh S, et al. GW100: benchmarking G0W0 for molecular systems. J Chem Theory Comput. 2015 Oct;11:5665–5687.
  • Liu P, Kaltak M, Klimeš J, et al. Cubic scaling GW: towards fast quasiparticle calculations. Phys Rev B. 2016 Oct;94:165109.
  • Wilhelm J, Golze D, Talirz L, et al. Toward GW calculations on thousands of atoms. J Phys Chem Lett. 2018 Jan;9:306–312.
  • Hedin L. On correlation effects in electron spectroscopies and the GW approximation. J Phys Condens Matter. 1999 Oct;11:R489.
  • Van Setten MJ, Weigend F, Evers F. The GW-method for quantum chemistry applications: theory and implementation. J Chem Theory Comput. 2013 Jan;9:232–246.
  • Bruneval F, Rangel T, Hamed SM, et al. MOLGW 1: many-body perturbation theory software for atoms, molecules, and clusters. Comput Phys Commun. 2016 Nov;208:149–161.
  • Duchemin I, Blase X. Robust analytic-continuation approach to many-body GW calculations. J Chem Theory Comput. 2020 Mar;16:1742–1756.
  • Salpeter EE, Bethe HA. A relativistic equation for bound-state problems. Phys Rev. 1951 Dec;84:1232–1242.
  • Fetter A, Walecka J. Quantum theory of many-particle systems. Dover Books on Physics. Dover: Dover Publications, 2012.
  • Jacquemin D, Duchemin I, Blondel A, et al. Benchmark of bethe-Salpeter for triplet excited-states. J Chem Theory Comput. 2017 Feb;13:767–783.
  • Rangel T, Rinn A, Sharifzadeh S, et al. Low-lying excited states in crystalline perylene. Proc Natl Acad Sci U S A. 2017 Jan;115:284–289.
  • Baumeier B, Andrienko D, Ma Y, et al. Excited states of dicyanovinyl-substituted oligothiophenes from many-body Green’s functions theory. J Chem Theory Comput. 2012 Mar;8:997–1002.
  • Loos PF, Blase X. Dynamical correction to the Bethe-Salpeter equation beyond the plasmon-pole approximation. J Chem Phys. 2020 Sep;153:114120.
  • Duchemin I, Deutsch T, Blase X. Short-range to long-range charge-transfer excitations in the zincbacteriochlorin-bacteriochlorin complex: a bethe-salpeter study. Phys Rev Lett. 2012 Oct;109:167801.
  • Van Der Horst JW, Bobbert PA, Michels MA, et al. Calculation of excitonic properties of conjugated polymers using the Bethe-Salpeter equation. J Chem Phys. 2001 Apr;114:6950–6957.
  • Hummer K, Puschnig P, Ambrosch-Draxl C. Lowest optical excitations in molecular crystals: bound excitons versus free electron-hole pairs in anthracene. Phys Rev Lett. 2004 Apr;92:147402.
  • Prezzi D, Varsano D, Ruini A, et al. Optical properties of graphene nanoribbons: the role of many-body effects. Phys Rev B. 2008 Jan;77:041404.
  • Blase X, Attaccalite C. Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach. Appl Phys Lett. 2011 Oct;99:171909.
  • Rocca D, Lu D, Galli G. Ab initio calculations of optical absorption spectra: solution of the Bethe-Salpeter equation within density matrix perturbation theory. J Chem Phys. 2010 Oct;133:164109.
  • Baumeier B, Andrienko D, Rohlfing M. Frenkel and charge-transfer excitations in donor-acceptor complexes from many-body green’s functions theory. J Chem Theory Comput. 2012 Aug;8:2790–2795.
  • Cudazzo P, Gatti M, Rubio A. Excitons in molecular crystals from first-principles many-body perturbation theory: picene versus pentacene. Phys Rev Lett. 2012 Nov;86:195307.
  • Sharifzadeh S, Darancet P, Kronik L, et al. Low-energy charge-transfer excitons in organic solids from first-principles: the case of pentacene. J Phys Chem Lett. 2013 Jul;4:2197–2201.
  • Qiu DY, Da Jornada FH, Louie SG. Optical spectrum of MoS2: many-body effects and diversity of exciton states. Phys Rev Lett. 2013 Nov;111:216805.
  • Yin H, Ma Y, Mu J, et al. Charge-transfer excited states in aqueous DNA: insights from many-body green’s function theory. Phys Rev Lett. 2014 Jun;112:228301.
  • Li Y, Shu H, Wang S, et al. Electronic and optical properties of graphene quantum dots: the role of many-body effects. J Phys Chem C. 2015;119:4983–4989.
  • Aggoune W, Cocchi C, Nabok D, et al. Dimensionality of excitons in stacked van der Waals materials: the example of hexagonal boron nitride. Phys Rev B. 2018 Jun;97:241114.
  • Valencia AM, Cocchi C. Electronic and optical properties of oligothiophene-f4tcnq charge-transfer complexes: the role of the donor conjugation length. J Phys Chem C. 2019;123:9617–9623.
  • Vorwerk C, Aurich B, Cocchi C, et al. Bethe–Salpeter equation for absorption and scattering spectroscopy: implementation in the exciting code. Science. 2019 Aug;1:037001.
  • Ismail-Beigi S, Louie SG. Excited-state forces within a first-principles Green’s function formalism. Phys Rev Lett. 2003 Feb;90:4.
  • Çaylak O, Baumeier B. Excited-state geometry optimization of small molecules with many-body green’s functions theory. J Chem Theory Comput. 2021;17:879–888.
  • Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev Lett. 1996 Oct;54:11169–11186.
  • Deslippe J, Samsonidze G, Strubbe DA, et al. Berkeleygw: a massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Comput Phys Commun. 2012;183:1269–1289.
  • Sangalli D, Ferretti A, Miranda H, et al. Many-body perturbation theory calculations using the yambo code. J Phys Condens Matter. 2019 May;31:325902.
  • Kittel C. Introduction to solid state physics. 8th ed. Weinheim: Wiley; 2004.
  • Kühne TD, Iannuzzi M, Del Ben M, et al. CP2K: an electronic structure and molecular dynamics software package -Quickstep: efficient and accurate electronic structure calculations. J Chem Phys. 2020 May;152:194103.
  • Vorwerk C, Cocchi C, Draxl C. Addressing electron-hole correlation in core excitations of solids: an all-electron many-body approach from first principles. Phys Rev B. 2017 Apr;95:155121.
  • Rohlfing M. Electronic excitations from a perturbative LDA+GdW approach. Phys Rev Lett. 2010 Nov;82:205127.
  • Marsili M, Mosconi E, De Angelis F, et al. Large-scale GW -BSE calculations with N3 scaling: excitonic effects in dye-sensitized solar cells. Phys Rev B. 2017 Feb;95:075415.
  • Bokdam M, Sander T, Stroppa A, et al. Role of polar phonons in the photo excited state of metal Halide Perovskites. Sci Rep. 2016 Jun;6:1–8.
  • Dvorak M, Golze D, Rinke P. Quantum embedding theory in the screened Coulomb interaction: combining configuration interaction with GW/BSE. Phys Rev Mater. 2019 Jul;3:070801.
  • Manby FR, Stella M, Goodpaster JD, et al. A simple, exact density-functional-theory embedding scheme. J Chem Theory Comput. 2012 Aug;8:2564–2568.
  • Petras HR, Graham DS, Ramadugu SK, et al. Fully quantum embedding with density functional theory for full configuration interaction quantum Monte Carlo. J Chem Theory Comput. 2019 Oct;15:5332–5342.
  • Wen X, Graham DS, Chulhai DV, et al. Absolutely localized projection-based embedding for excited states. J Chem Theory Comput. 2020 Jan;16:385–398.
  • Li J, D’Avino G, Pershin A, et al. Correlated electron-hole mechanism for molecular doping in organic semiconductors. Phys Rev Mater. 2017 Jul;1:25602.
  • Li J, D’Avino G, Duchemin I, et al. Accurate description of charged excitations in molecular solids from embedded many-body perturbation theory. Phys Rev B. 2018 Jan;97:35108.
  • Tirimbò G, Sundaram V, Çaylak O, et al. Excited-state electronic structure of molecules using many-body Green’s functions: quasiparticles and electron-hole excitations with VOTCA-XTP. J Chem Phys. 2020 Mar;152:114103.
  • Duchemin I, Guido CA, Jacquemin D, et al. The Bethe-Salpeter formalism with polarisable continuum embedding: reconciling linear-response and state-specific features. Chem Sci. 2018 May;9:4430–4443.
  • Duchemin I, Jacquemin D, Blase X. Combining the GW formalism with the polarizable continuum model: a state-specific non-equilibrium approach. J Chem Phys. 2016 Apr;144:164106.
  • Wehner J, Brombacher L, Brown J, et al. Electronic excitations in complex molecular environments: many-Body Green’s functions theory in VOTCA-XTP. J Chem Theory Comput. 2018 Dec;14:6253–6268.
  • Van Duijnen PT, Swart M. Molecular and atomic polarizabilities: thole’s model revisited. J Phys Chem A. 1998 Apr;102:2399–2407.
  • Fitzner R, Mena-Osteritz E, Mishra A, et al. Correlation of π -conjugated oligomer structure with film morphology and organic solar cell performance. J Am Chem Soc. 2012 Jul;134:11064–11067.
  • Poelking C, Tietze M, Elschner C, et al. Impact of mesoscale order on open-circuit voltage in organic solar cells. Nat Mater. 2015 Dec;14:434–439.
  • Banyasz A, Vayá I, Changenet-Barret P, et al. Base pairing enhances fluorescence and favors cyclobutane dimer formation induced upon absorption of UVA radiation by DNA. J Am Chem Soc. 2011 Apr;133:5163–5165.
  • Lobo BC, Abelt CJ. Does PRODAN possess a planar or twisted charge-transfer excited state? Photophysical properties of two PRODAN derivatives. J Phys Chem A. 2003 Dec;107:10938–10943.
  • Baral S, Phillips M, Yan H, et al. Ultrafast formation of the charge transfer state of prodan reveals unique aspects of the chromophore environment. J Phys Chem B. 2020 Apr;124:2643–2651.
  • Sanchez SA, Tricerri MA, Gratton E. Laurdan generalized polarization fluctuations measures membrane packing micro-heterogeneity in vivo. Proc Natl Acad Sci U S A. 2012 May;109:7314–7319.
  • Masukawa MK, Vequi-Suplicy CC, Duarte EL, et al. A closer look into laurdan as a probe to monitor cationic DODAB bilayers. J Photochem Photobiol A. 2019 May;376:238–246.
  • Wang J, Cieplak P, Luo R, et al. Development of polarizable gaussian model for molecular mechanical calculations i: atomic polarizability parameterization to reproduce ab initio anisotropy. J Chem Theory Comput. 2019;15:1146–1158.
  • Nistor RA, Polihronov JG, Müser MH, et al. A generalization of the charge equilibration method for nonmetallic materials. The Journal of Chemical Physics. 2006 Sep;125:094108.
  • Mathieu D. Split charge equilibration method with correct dissociation limits. J Chem Phys. 2007 Dec;127:224103.