Advanced Review
On the top rung of Jacob's ladder of density functional theory: Toward resolving the dilemma of SIE and NCE
Igor Ying Zhang,
Xin Xu,
Corresponding Author
Igor Ying Zhang
Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai, China
Correspondence
Igor Ying Zhang and Xin Xu, Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai 200433, China. Email: [email protected] (I. Y. Z.) and [email protected] (X. X.)
Search for more papers by this authorCorresponding Author
Xin Xu
Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai, China
Correspondence
Igor Ying Zhang and Xin Xu, Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai 200433, China. Email: [email protected] (I. Y. Z.) and [email protected] (X. X.)
Search for more papers by this authorIgor Ying Zhang,
Xin Xu,
Corresponding Author
Igor Ying Zhang
Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai, China
Correspondence
Igor Ying Zhang and Xin Xu, Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai 200433, China. Email: [email protected] (I. Y. Z.) and [email protected] (X. X.)
Search for more papers by this authorCorresponding Author
Xin Xu
Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai, China
Correspondence
Igor Ying Zhang and Xin Xu, Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai 200433, China. Email: [email protected] (I. Y. Z.) and [email protected] (X. X.)
Search for more papers by this authorFunding information: National Key Research and Development Program of China, Grant/Award Number: 2018YFA0208600; National Natural Science Foundation of China, Grant/Award Numbers: 21688102, 21973015; Science Challenge Project, Grant/Award Number: TZ2018004
Abstract
According to the classification of Jacob's Ladder proposed by Perdew, density functional approximations (DFAs) on the top (fifth) rung add the information of the unoccupied Kohn–Sham orbitals, which hold the promise to enter the heaven of chemical accuracy. In other words, we expect that a much higher accuracy with a broader applicability than the existing DFAs would eventually be achieved on the fifth rung. Nonetheless, Jacob's Ladder itself does not offer a recipe for how to manipulate the unoccupied Kohn–Sham orbitals on the construction of a successful fifth rung DFA. In this article, we briefly review two successful types of the fifth rung DFAs, that is, random-phase approximation (RPA) and doubly hybrid approximation (DHA). The limitations of RPA and DHA will be introduced in the context of the so-called self-interaction error (SIE)/nondynamic correlation error (NCE) dilemma in the world of density functional theory. We propose the development strategy for DHAs to address the general concern about the future of advanced DFAs on the fifth rung. We share our experience here, based on the relevant efforts recently made by the authors and their co-workers, aiming to resolve the SIE/NCE dilemma and to extend the applicability of DHAs from the chemistry of the main group elements to that of the transition metals.
This article is categorized under:
- Electronic Structure Theory > Density Functional Theory
- Software > Quantum Chemistry
Graphical Abstract
Jacob's Ladder of DFT towards Chemical Accuracy.
CONFLICT OF INTEREST
The authors have declared no conflicts of interest for this article.
REFERENCES
- 1Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys Rev. 1965; 140(4A): A1133–A1138.
- 2Levy M, Yang W, Parr RG. A new functional with homogeneous coordinate scaling in density functional theory: F [ ρ,λ]. J Chem Phys. 1985; 83(5): 2334–2336.
- 3Becke AD. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A. 1988; 38(6): 3098–3100.
- 4Becke AD. A new mixing of Hartree–Fock and local density-functional theories. J Chem Phys. 1993; 98(2): 1372–1377.
- 5Becke AD. Density-functional thermochemistry 3: The role of exact exchange. J Chem Phys. 1993; 98(7): 5648–5652.
- 6Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys Rev Lett. 1996; 77(18): 3865–3868.
- 7Perdew JP, Emzerhof M, Burke K. Rationale for mixing exact exchange with density functional approximations. J Chem Phys. 1996; 105(22): 9982–9985.
- 8Seidl A, Görling A, Vogl P, Majewski JA, Levy M. Generalized Kohn-Sham schemes and the band-gap problem. Phys Rev B. 1996; 53(7): 3764–3774.
- 9Becke AD. Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals. J Chem Phys. 1997; 107(20): 8554–8560.
- 10Ernzerhof M, Scuseria GE. Assessment of the Perdew–Burke–Ernzerhof exchange-correlation functional. J Chem Phys. 1999; 110(11): 5029–5036.
- 11Carlo A, Vincenzo B. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J Chem Phys. 1999; 110(13): 6158–6170.
- 12Tao JM, Perdew JP, Staroverov VN, Scuseria GE. Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids. Phys Rev Lett. 2003; 91(14): 146401–146404.
- 13Heyd J, Scuseria G, Ernzerhof M. Hybrid functionals based on a screened coulomb potential. J Chem Phys. 2003; 118(18): 8207–8215.
- 14Xu X, Goddard WA III. The X3LYP extended density functional for accurate descriptions of nonbond interactions, spin states, and thermochemical properties. Proc Natl Acad Sci USA. 2004; 101(9): 2673–2677.
- 15Xu X, Goddard WA III. The extended Perdew-Burke-Ernzerhof functional with improved accuracy for thermodynamic and electronic properties of molecular systems. J Chem Phys. 2004; 121(9): 4068–4082.
- 16Zhao Y, Truhlar DG. A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J Chem Phys. 2006; 125(19): 194101–194118.
- 17Chai J-D, Head-Gordon M. Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections. Phys Chem Chem Phys. 2008; 10(44): 6615–6620.
- 18Zhao Y, Truhlar DG. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: Two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor Chem Acc. 2008; 120(1–3): 215–241.
- 19Sun J, Ruzsinszky A, Perdew JP. Strongly constrained and appropriately normed semilocal density functional. Phys Rev Lett. 2015; 115(3):036402.
- 20Mardirossian N, Head-Gordon M. Thirty years of density functional theory in computational chemistry: An overview and extensive assessment of 200 density functionals. Mol Phys. 2017; 115(19): 2315–2372.
- 21Goerigk L, Mehta N. A trip to the density functional theory zoo: Warnings and recommendations for the user. Aust J Chem. 2019; 72(8): 563–573.
- 22Curtiss LA, Raghavachari K, Redfern PC, Pople JA. Assessment of Gaussian-3 and density functional theories for a larger experimental test set. J Chem Phys. 2000; 112(17): 7374–7383.
- 23Wodrich MD, Corminboeuf C, Schleyer PV. Systematic errors in computed alkane energies using B3LYP and other popular DFT functionals. Org Lett. 2006; 8(17): 3631–3634.
- 24Zhang IY, Wu JM, Xu X. Extending the reliability and applicability of B3LYP. Chem Commun. 2010; 46(18): 3057–3070.
- 25Zhao Y, Truhlar DG. Benchmark databases for nonbonded interactions and their use to test density functional theory. J Chem Theory Comput. 2005; 1(3): 415–432.
- 26Langreth DC, Perdew JP. Exchange-correlation energy of a metallic surface: Wave-vector analysis. Phys Rev B. 1977; 15(6): 2884–2901.
- 27Furche F. Molecular tests of the random phase approximation to the exchange-correlation energy functional. Phys Rev B. 2001; 64(19): 195120–195128.
- 28Furche F, van Voorhis T. Fluctuation-dissipation theorem density-functional theory. J Chem Phys. 2005; 122(16): 164106–164110.
- 29Harl J, Kresse G. Cohesive energy curves for noble gas solids calculated by adiabatic connection fluctuation-dissipation theory. Phys Rev B. 2008; 77(4):045136.
- 30Fuchs M, Gonze X. Accurate density functionals: Approaches using the adiabatic-connection fluctuation-dissipation theorem. Phys Rev B. 2002; 65(23):235109.
- 31Grimme S. Semiempirical hybrid density functional with perturbative second-order correlation. J Chem Phys. 2006; 124(3):034108.
- 32Tarnopolsky A, Karton A, Sertchook R, Vuzman D, Martin JML. Double-hybrid functionals for thermochemical kinetics. Chem A Eur J. 2008; 112(1): 3–8.
- 33Karton A, Tarnopolsky A, Lamère J-F, Schatz GC, Martin JML. Highly accurate first-principles benchmark data sets for the parametrization and validation of density functional and other approximate methods. Derivation of a robust, generally applicable, double-hybrid functional for thermochemistry and thermochemical kinetics. Chem A Eur J. 2008; 112(50): 12868–12886.
- 34Chai JD, Head-Gordon M. Long-range corrected double-hybrid density functionals. J Chem Phys. 2009; 131(17): 174105–174113.
- 35Zhang Y, Xu X, Goddard WA III. Doubly hybrid density functional for accurate descriptions of nonbond interactions, thermochemistry, and thermochemical kinetics. Proc Natl Acad Sci USA. 2009; 106(13): 4963–4968.
- 36Kozuch S, Gruzman D, Martin JML. DSD-BLYP: A general purpose double hybrid density functional including spin component scaling and dispersion correction. J Phys Chem C. 2010; 114(48): 20801–20808.
- 37Goerigk L, Grimme S. Efficient and accurate double-hybrid-meta-GGA density functionals—Evaluation with the extended GMTKN30 database for general main group thermochemistry, kinetics, and noncovalent interactions. J Chem Theory Comput. 2011; 7(2): 291–309.
- 38Kozuch S, Martin JML. DSD-PBEP86: In search of the best double-hybrid DFT with spin-component scaled MP2 and dispersion corrections. Phys Chem Chem Phys. 2011; 13(45): 20104–20107.
- 39Sharkas K, Toulouse J, Savin A. Double-hybrid density-functional theory made rigorous. J Chem Phys. 2011; 134(6):064113.
- 40Zhang IY, Xu X, Jung Y, Goddard WA III. A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz. Proc Natl Acad Sci USA. 2011; 108(50): 19896–19900.
- 41Zhang IY, Su NQ, Brémond ÉAG, Adamo C, Xu X. Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0. J Chem Phys. 2012; 136(17): 174103–174108.
- 42Zhang IY, Xu X. Reaching a uniform accuracy for complex molecular systems: Long-range-corrected XYG3 doubly hybrid density functional. J Phys Chem Lett. 2013; 4(10): 1669–1675.
- 43Brémond E, Adamo C. Seeking for parameter-free double-hybrid functionals: The PBE0-DH model. J Chem Phys. 2011; 135(2):024106.
- 44Brémond É, Sancho-García JC, Pérez-Jiménez ÁJ, Adamo C. Double-hybrid functionals from adiabatic-connection: The QIDH model. J Chem Phys. 2014; 141(3):031101.
- 45Su NQ, Xu X. Construction of a parameter-free doubly hybrid density functional from adiabatic connection. J Chem Phys. 2014; 140(18): 18A512–18A515.
- 46Brémond É, Savarese M, Pérez-Jiménez ÁJ, Sancho-García JC, Adamo C. Range-separated double-hybrid functional from nonempirical constraints. J Chem Theory Comput. 2018; 14(8): 4052–4062.
- 47Mardirossian N, Head-Gordon M. Survival of the most transferable at the top of Jacob's ladder: Defining and testing the omega B97M(2) double hybrid density functional. J Chem Phys. 2018; 148(24):241736.
- 48Ren X, Rinke P, Joas C, Scheffler M. Random-phase approximation and its applications in computational chemistry and materials science. J Mater Sci. 2012; 47(21): 7447–7471.
- 49Eshuis H, Bates J, Furche F. Electron correlation methods based on the random phase approximation. Theor Chem Acc. 2012; 131(1): 1084.
- 50Chen GP, Voora VK, Agee MM, Balasubramani SG, Furche F. Random-phase approximation methods. Annu Rev Phys Chem. 2017; 68(1): 421–445.
- 51Zhang IY, Xu X. Doubly hybrid density functional for accurate description of thermochemistry, thermochemical kinetics and nonbonded interactions. Int Rev Phys Chem. 2011; 30(1): 115–160.
- 52Sancho-García JC, Adamo C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds. Phys Chem Chem Phys. 2013; 15(35): 14581–14594.
- 53Goerigk L, Grimme S. Double-hybrid density functionals. WIREs Comput Mol Sci. 2014; 4(6): 576–600.
- 54Zhang IY, Xu X. A new generation of doubly hybrid density functionals (DHDFs). A new-generation density functional. Berlin/Heidelberg: Springer, 2014; p. 25–45.
10.1007/978-3-642-40421-4_2 Google Scholar
- 55Su NQ, Xu X. The XYG3 type of doubly hybrid density functionals. WIREs Comput Mol Sci. 2016; 6(6): 721–747.
- 56Martin JML, Santra G. Empirical double-hybrid density functional theory: A ‘third way’ in between WFT and DFT. Israel J Chem. 2020; 60: 1–19.
- 57Whitten JL. Coulombic potential energy integrals and approximations. J Chem Phys. 1973; 58(10): 4496–4501.
- 58Weigend F, Häser M, Patzelt H, Ahlrichs R. RI-MP2: Optimized auxiliary basis sets and demonstration of efficiency. Chem Phys Lett. 1998; 294(1): 143–152.
- 59Dunlap BI. Robust variational fitting: Gáspár's variational exchange can accurately be treated analytically. J Mol Struct (THEOCHEM). 2000; 501–502: 221–228.
- 60Jung Y, Sodt A, Gill PMW, Head-Gordon M. Auxiliary basis expansions for large-scale electronic structure calculations. Proc Natl Acad Sci USA. 2005; 102(19): 6692–6697.
- 61Dunlap BI, Rösch N, Trickey SB. Variational fitting methods for electronic structure calculations. Mol Phys. 2010; 108(21–23): 3167–3180.
- 62Ren X, Rinke P, Blum V, et al. Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions. New J Phys. 2012; 14:053020-60.
- 63Head-Gordon M. An efficient implementation of the pair atomic resolution of the identity approximation for exact exchange for hybrid and range-separated density functionals. J Chem Theory Comput. 2014; 11(2): 518–524.
- 64Ihrig AC, Wieferink J, Zhang IY, et al. Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory. New J Phys. 2015; 17(9):093020.
- 65Häser M, Almlöf J. Laplace transform techniques in Møller–Plesset perturbation theory. J Chem Phys. 1992; 96(1): 489–494.
- 66Ayala PY, Scuseria GE. Linear scaling second-order Møller–Plesset theory in the atomic orbital basis for large molecular systems. J Chem Phys. 1999; 110(8): 3660–3671.
- 67Schütz M, Hetzer G, Werner H-J. Low-order scaling local electron correlation methods. I. Linear scaling local MP2. J Chem Phys. 1999; 111(13): 5691–5705.
- 68Jung Y, Shao Y, Head-Gordon M. Fast evaluation of scaled opposite spin second-order Møller–Plesset correlation energies using auxiliary basis expansions and exploiting sparsity. J Comput Chem. 2007; 28(12): 1953–1964.
- 69Yang J, Kurashige Y, Manby FR, Chan GKL. Tensor factorizations of local second-order Møller–Plesset theory. J Chem Phys. 2011; 134(4):044123.
- 70Schmitz G, Helmich B, Hättig C. A scaling PNO–MP2 method using a hybrid OSV–PNO approach with an iterative direct generation of OSVs. Mol Phys. 2013; 111(16–17): 2463–2476.
- 71Pinski P, Riplinger C, Valeev EF, Neese F. Sparse maps—A systematic infrastructure for reduced-scaling electronic structure methods. I. an efficient and simple linear scaling local MP2 method that uses an intermediate basis of pair natural orbitals. J Chem Phys. 2015; 143(3):034108.
- 72Kjærgaard T. The Laplace transformed divide-expand-consolidate resolution of the identity second-order Møller-Plesset perturbation (DEC-LT-RIMP2) theory method. J Chem Phys. 2017; 146(4):044103.
- 73Kaltak M, Klimeš J, Kresse G. Cubic scaling algorithm for the random phase approximation: Self-interstitials and vacancies in Si. Phys Rev B. 2014; 90(5):054115.
- 74Kállay M. Linear-scaling implementation of the direct random-phase approximation. J Chem Phys. 2015; 142(20):204105.
- 75Maurer SA, Clin L, Ochsenfeld C. Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems. J Chem Phys. 2014; 140(22):224112.
- 76Perdew, J. P.; Schmidt, K. (2000). Jacob's ladder of density functional approximations for the exchange-correlation energy. In V. Doren (Ed.), Density functional theory and its application to materials, Vol. 577. College Park, MD: American Institute of Physics.
- 77Perdew JP. Climbing the ladder of density functional approximations. MRS Bull. 2013; 38(09): 743–750.
- 78Paier J. Hybrid density functionals applied to complex solid catalysts: Successes, limitations, and prospects. Catal Lett. 2016; 146(5): 861–885.
- 79Maurer RJ, Freysoldt C, Reilly AM, et al. Advances in density-functional calculations for materials modeling. Annu Rev Mat Res. 2019; 49(1): 1–30.
- 80Mori-Sánchez P, Cohen AJ, Yang W. Failure of the random-phase-approximation correlation energy. Phys Rev A. 2012; 85(4):042507.
- 81Grüneis A, Marsman M, Harl J, Schimka L, Kresse G. Making the random phase approximation to electronic correlation accurate. J Chem Phys. 2009; 131(15): 154115–154115.
- 82Ren X, Tkatchenko A, Rinke P, Scheffler M. Beyond the random-phase approximation for the electron correlation energy: The importance of single excitations. Phys Rev Lett. 2011; 106(15): 153003–153004.
- 83Ren X, Rinke P, Scuseria GE, Scheffler M. Renormalized second-order perturbation theory for the electron correlation energy: Concept, implementation, and benchmarks. Phys Rev B. 2013; 88(3):035120.
- 84Scuseria GE, Henderson TM, Bulik IW. Particle-particle and quasiparticle random phase approximations: Connections to coupled cluster theory. J Chem Phys. 2013; 139(10):104113.
- 85Aggelen, H. V., Yang, Y., Yang, W.. Exchange-correlation energy from pairing matrix fluctuation and the particle-particle random phase approximation. J Chem Phys. 2014, 140(18): 18A511.
- 86Bates JE, Furche F. Random phase approximation renormalized many-body perturbation theory. J Chem Phys. 2013; 139(17):171103.
- 87Heßelmann A, Görling A. Random phase approximation correlation energies with exact Kohn–Sham exchange. Mol Phys. 2010; 108(3–4): 359–372.
- 88Görling A. Hierarchies of methods towards the exact Kohn-Sham correlation energy based on the adiabatic-connection fluctuation-dissipation theorem. Phys Rev B. 2019; 99(23):235120.
- 89Bates JE, Laricchia S, Ruzsinszky A. Nonlocal energy-optimized kernel: Recovering second-order exchange in the homogeneous electron gas. Phys Rev B. 2016; 93(4):045119.
- 90Mezei PD, Csonka GI, Ruzsinszky A, Kállay M. Construction and application of a new dual-hybrid random phase approximation. J Chem Theory Comput. 2015; 11(10): 4615–4626.
- 91Grimme S, Steinmetz M. A computationally efficient double hybrid density functional based on the random phase approximation. Phys Chem Chem Phys. 2016; 18(31): 20926–20937.
- 92Mezei PD, Csonka GI, Ruzsinszky A, Kállay M. Construction of a spin-component scaled dual-hybrid random phase approximation. J Chem Theory Comput. 2017; 13(2): 796–803.
- 93Chan B, Goerigk L, Radom L. On the inclusion of post-MP2 contributions to double-hybrid density functionals. J Comput Chem. 2016; 37(2): 183–193.
- 94Tran F, Stelzl J, Blaha P. Rungs 1 to 4 of DFT Jacob's ladder: Extensive test on the lattice constant, bulk modulus, and cohesive energy of solids. J Chem Phys. 2016; 144(20):204120.
- 95Perdew JP, Parr RG, Levy M, Balduz JL. Density-functional theory for fractional particle number: Derivative discontinuities of the energy. Phys Rev Lett. 1982; 49(23): 1691–1694.
- 96Görling A, Levy M. Correlation-energy functional and its high-density limit obtained from a coupling-constant perturbation expansion. Phys Rev B. 1993; 47(20): 13105–13113.
- 97Levy M, Perdew JP. Hellmann–Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. Phys Rev A. 1985; 32(4): 2010–2021.
- 98Su NQ, Xu X. Development of new density functional approximations. Annu Rev Phys Chem. 2017; 68(1): 155–182.
- 99Becke AD. Perspective: Fifty years of density-functional theory in chemical physics. J Chem Phys. 2014; 140(18):18A301.
- 100Zhang IY, Rinke P, Perdew JP, Scheffler M. Towards efficient orbital-dependent density functionals for weak and strong correlation. Phys Rev Lett. 2016; 117(13):133002.
- 101Zhang IY, Rinke P, Scheffler M. Wave-function inspired density functional applied to the H2/H2+ challenge. New J Phys. 2016; 18(7):073026.
- 102Zhang IY, Xu X. Simultaneous attenuation of both self-interaction error and nondynamic correlation error in density functional theory: A spin-pair distinctive adiabatic-connection approximation. J Phys Chem Lett. 2019; 10: 2617–2623.
- 103Perdew JP, Zunger A. Self-interaction correction to density-functional approximations for many-electron systems. Phys Rev B. 1981; 23(10): 5048–5079.
- 104Cohen AJ, Mori-Sánchez P, Yang W. Challenges for density functional theory. Chem Rev. 2011; 112(1): 289–320.
- 105Zhang Y, Yang W. A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons. J Chem Phys. 1998; 109(7): 2604–2608.
- 106Perdew JP, Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B. 1992; 45(23): 13244–13249.
- 107Ruzsinszky A, Zhang IY, Scheffler M. Insight into organic reactions from the direct random phase approximation and its corrections. J Chem Phys. 2015; 143(14):144115.
- 108Zhang IY, Ren X, Rinke P, Blum V, Scheffler M. Numeric atom-centered-orbital basis sets with valence-correlation consistency from H to Ar. New J Phys. 2013; 15(12):123033.
- 109Shen T, Zhu Z, Zhang IY, Scheffler M. Massive-parallel implementation of the resolution-of-identity coupled-cluster approaches in the numeric atom-centered orbital framework for molecular systems. J Chem Theory Comput. 2019; 15(9): 4721–4734.
- 110Blum V, Gehrke R, Hanke F, et al. Ab initio molecular simulations with numeric atom-centered orbitals. Comput Phys Commun. 2009; 180(11): 2175–2196.
- 111Mori-Sánchez P, Cohen AJ, Yang WT. Many-electron self-interaction error in approximate density functionals. J Chem Phys. 2006; 125(20):201102.
- 112Ruzsinszky A, Perdew JP, Csonka GI, Vydrov OA, Scuseria GE. Density functionals that are one- and two- are not always many-electron self-interaction-free, as shown for H2+, He2+, LiH+, and Ne2+. J Chem Phys. 2007; 126(10):104102.
- 113Ruzsinszky A, Perdew JP, Csonka GI, Vydrov OA, Scuseria GE. Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals. J Chem Phys. 2006; 125(19):194112.
- 114Vydrov OA, Scuseria GE, Perdew JP, Ruzsinszky A, Csonka GI. Scaling down the Perdew-Zunger self-interaction correction in many-electron regions. J Chem Phys. 2006; 124(9):094108.
- 115Cohen AJ, Mori-Sanchez P, Yang WT. Insights into current limitations of density functional theory. Science. 2008; 321(5890): 792–794.
- 116Li C, Yang W. On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals. J Chem Phys. 2017; 146(7):074107.
- 117Mori-Sánchez P, Cohen AJ, Yang W. Localization and delocalization errors in density functional theory and implications for band-gap prediction. Phys Rev Lett. 2008; 100(14):146401.
- 118Booth GH, Thom AJW, Alavi A. Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in slater determinant space. J Chem Phys. 2009; 131(5):054106.
- 119Ruzsinszky A, Perdew JP. Twelve outstanding problems in ground-state density functional theory: A bouquet of puzzles. Comput Theor Chem. 2011; 963(1): 2–6.
- 120Sharkas K, Savin A, Jensen HJA, Toulouse J. A multiconfigurational hybrid density-functional theory. J Chem Phys. 2012; 137(4):044104.
- 121Li Manni G, Carlson RK, Luo S, et al. Multi-configuration pair-density functional theory. J Chem Theory Comput. 2014; 10: 3669–3680.
- 122Chen Z, Zhang D, Jin Y, Yang Y, Su NQ, Yang W. Multireference density functional theory with generalized auxiliary systems for ground and excited states. J Phys Chem Lett. 2017; 8(18): 4479–4485.
- 123Mori-Sánchez P, Cohen AJ, Yang W. Discontinuous nature of the exchange-correlation functional in strongly correlated systems. Phys Rev Lett. 2009; 102(6):066403.
- 124Su NQ, Li C, Yang W. Describing strong correlation with fractional-spin correction in density functional theory. Proc Natl Acad Sci USA. 2018; 115(39): 9678–9683.
- 125Li C, Zheng X, Su NQ, Yang W. Localized orbital scaling correction for systematic elimination of delocalization error in density functional approximations. Natl Sci Rev. 2018; 5(2): 203–215.
- 126Su NQ, Yang W, Mori-Sánchez P, Xu X. Fractional charge behavior and band gap predictions with the xyg3 type of doubly hybrid density functionals. Chem A Eur J. 2014; 118(39): 9201–9211.
- 127Szabo A, Ostlund NS. Modern quantum chemistry. New York, NY: McGraw-Hill, 1996.
- 128Tkatchenko A, Scheffler M. Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data. Phys Rev Lett. 2009; 102(7):073005.
- 129Pritchard BP, Altarawy D, Didier B, Gibson TD, Windus TL. New basis set exchange: An open, up-to-date resource for the molecular sciences community. J Chem Inf Model. 2019; 59(11): 4814–4820.
- 130HeatonBurgess T, Bulat FA, Yang W. Optimized effective potentials in finite basis sets. Phys Rev Lett. 2007; 98(25):256401.
- 131Smiga S, Franck O, Mussard B, et al. Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method. J Chem Phys. 2016; 145(14):144102.
- 132Neese F, Schwabe T, Kossmann S, Schirmer B, Grimme S. Assessment of orbital-optimized, spin-component scaled second-order many-body perturbation theory for thermochemistry and kinetics. J Chem Theory Comput. 2009; 5(11): 3060–3073.
- 133Stueck D, Head-Gordon M. Regularized orbital-optimized second-order perturbation theory. J Chem Phys. 2013; 139(24):244109.
- 134Atalla V, Zhang IY, Hofmann OT, Ren X, Rinke P, Scheffler M. Enforcing the linear behavior of the total energy with hybrid functionals: Implications for charge transfer, interaction energies, and the random-phase approximation. Phys Rev B. 2016; 94(3):035104.
- 135Yang W, Mori-Sanchez P, Cohen AJ. Extension of many-body theory and approximate density functionals to fractional charges and fractional spins. J Chem Phys. 2013; 139(10):104114.
- 136Cheng L, Gauss J, Ruscic B, Armentrout PB, Stanton JF. Bond dissociation energies for diatomic molecules containing 3d transition metals: Benchmark scalar-relativistic coupled-cluster calculations for 20 molecules. J Chem Theory Comput. 2017; 13(3): 1044–1056.
- 137Aoto YA, de Lima Batista AP, Koehn A, de Oliveira-Filho AGS. How to arrive at accurate benchmark values for transition metal compounds: Computation or experiment? J Chem Theory Comput. 2017; 13(11): 5291–5316.
- 138Peverati R, Truhlar DG. Quest for a universal density functional: The accuracy of density functionals across a broad spectrum of databases in chemistry and physics. Phil Trans R Soc Lond A Math Phys Eng Sci. 2014; 372(2011):20120476.
- 139Zhao Y, Tishchenko O, Gour JR, et al. Thermochemical kinetics for multireference systems: Addition reactions of ozone. Chem A Eur J. 2009; 113(19): 5786–5799.
- 140Yu HS, He X, Li SL, Truhlar DG. MN15: A Kohn–Sham global-hybrid exchange–correlation density functional with broad accuracy for multi-reference and single-reference systems and noncovalent interactions. Chem Sci. 2016; 7(8): 5032–5051.
- 141Dohm S, Hansen A, Steinmetz M, Grimme S, Checinski MP. Comprehensive thermochemical benchmark set of realistic closed-shell metal organic reactions. J Chem Theory Comput. 2018; 14(5): 2596–2608.
- 142Barden CJ, Rienstra-Kiracofe JC, Schaefer HF. Homonuclear 3D transition-metal diatomics: A systematic density functional theory study. J Chem Phys. 2000; 113(2): 690–700.
- 143Schultz NE, Zhao Y, Truhlar DG. Databases for transition element bonding: Metal−metal bond energies and bond lengths and their use to test hybrid, hybrid meta, and meta density functionals and generalized gradient approximations. Chem A Eur J. 2005; 109(19): 4388–4403.
- 144Furche F, Perdew JP. The performance of semilocal and hybrid density functionals in 3D transition-metal chemistry. J Chem Phys. 2006; 124(4):044103.
- 145Lombardi JR, Davis B. Periodic properties of force constants of small transition-metal and lanthanide clusters. Chem Rev. 2002; 102(6): 2431–2460.
