Exchange-correlation functionals for band gaps of solids: benchmark, reparametrization and machine learning
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Abstract
Abstract We conducted a large-scale density-functional theory study on the influence of the exchange-correlation functional in the calculation of electronic band gaps of solids. First, we use the large materials data set that we have recently proposed to benchmark 21 different functionals, with a particular focus on approximations of the meta-generalized-gradient family. Combining these data with the results for 12 functionals in our previous work, we can analyze in detail the characteristics of each approximation and identify its strong and/or weak points. Beside confirming that mBJ, HLE16 and HSE06 are the most accurate functionals for band gap calculations, we reveal several other interesting functionals, chief among which are the local Slater potential approximation, the GGA AK13LDA, and the meta-GGAs HLE17 and TASK. We also compare the computational efficiency of these different approximations. Relying on these data, we investigate the potential for improvement of a promising subset of functionals by varying their internal parameters. The identified optimal parameters yield a family of functionals fitted for the calculation of band gaps. Finally, we demonstrate how to train machine learning models for accurate band gap prediction, using as input structural and composition data, as well as approximate band gaps obtained from density-functional theory.
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OpenAlex topics: Machine Learning in Materials Science · X-ray Diffraction in Crystallography · Advanced Thermoelectric Materials and Devices
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References (100+)
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1. Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).10.1103/PhysRev.136.B864 ↗
2. Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).10.1103/PhysRev.140.A1133 ↗
3. Cohen, A. J., Mori-Sánchez, P. & Yang, W. Challenges for density functional theory. Chem. Rev. 112, 289–320 (2012).10.1021/cr200107z ↗
4. Marques, M. A., Oliveira, M. J. & Burnus, T. Libxc: a library of exchange and correlation functionals for density functional theory. Comput. Phys. Commun. 183, 2272–2281 (2012).10.1016/j.cpc.2012.05.007 ↗
5. Lehtola, S., Steigemann, C., Oliveira, M. J. & Marques, M. A. Recent developments in libxc—a comprehensive library of functionals for density functional theory. SoftwareX 7, 1–5 (2018).10.1016/j.softx.2017.11.002 ↗
6. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).10.1103/PhysRevLett.77.3865 ↗
7. Perdew, J. P., Burke, K. & Ernzerhof, M. Erratum: generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)]. Phys. Rev. Lett. 78, 1396–1396 (1997).10.1103/PhysRevLett.78.1396 ↗
8. Becke, A. D. Density-functional thermochemistry. iii. the role of exact exchange. J. Chem. Phys. 98, 5648 (1993).10.1063/1.464913 ↗
9. Stephens, P. J., Devlin, F. J., Chabalowski, C. F. & Frisch, M. J. Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J. Phys. Chem. 98, 11623–11627 (1994).10.1021/j100096a001 ↗
10. Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened coulomb potential. J. Chem. Phys. 118, 8207 (2003).10.1063/1.1564060 ↗
11. Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: "Hybrid functionals based on a screened Coulomb potential” [J. Chem. Phys.118, 8207 (2003)]. J. Chem. Phys. 124, 219906 (2006).10.1063/1.2204597 ↗
12. Wiitala, K. W., Hoye, T. R. & Cramer, C. J. Hybrid density functional methods empirically optimized for the computation of 13C and 1H chemical shifts in chloroform solution. J. Chem. Theory Comput. 2, 1085–1092 (2006).10.1021/ct6001016 ↗
13. Jin, Y. & Bartlett, R. J. Accurate computation of X-ray absorption spectra with ionization potential optimized global hybrid functional. J. Chem. Phys. 149, 064111 (2018).10.1063/1.5038434 ↗
14. Sarmiento-Pérez, R., Botti, S. & Marques, M. A. L. Optimized exchange and correlation semilocal functional for the calculation of energies of formation. J. Chem. Theory Comput. 11, 3844–3850 (2015).10.1021/acs.jctc.5b00529 ↗
15. Tran, F. & Blaha, P. Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Phys. Rev. Lett. 102, 226401 (2009).10.1103/PhysRevLett.102.226401 ↗
16. Verma, P. & Truhlar, D. G. HLE16: a local Kohn–Sham gradient approximation with good performance for semiconductor band gaps and molecular excitation energies. J. Phys. Chem. Lett. 8, 380–387 (2017).10.1021/acs.jpclett.6b02757 ↗
17. Verma, P. & Truhlar, D. G. HLE17: an improved local exchange–correlation functional for computing semiconductor band gaps and molecular excitation energies. J. Phys. Chem. C. 121, 7144–7154 (2017).10.1021/acs.jpcc.7b01066 ↗
18. Sze, S. M. & Ng, K. K. Physics of Semiconductor Devices (Wiley, 2006).10.1002/0470068329 ↗
19. Neamen, D. A. Semiconductor Physics and Devices: Basic Principles (McGraw-Hill, New York, NY, 2012).
20. Perdew, J. P. & Levy, M. Physical content of the exact kohn-sham orbital energies: band gaps and derivative discontinuities. Phys. Rev. Lett. 51, 1884–1887 (1983).10.1103/PhysRevLett.51.1884 ↗
21. Sham, L. J. & Schlüter, M. Density-functional theory of the energy gap. Phys. Rev. Lett. 51, 1888–1891 (1983).10.1103/PhysRevLett.51.1888 ↗
22. Görling, A. Exchange-correlation potentials with proper discontinuities for physically meaningful kohn-sham eigenvalues and band structures. Phys. Rev. B 91, 245120 (2015).10.1103/PhysRevB.91.245120 ↗
23. Godby, R. W., Schlüter, M. & Sham, L. J. Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron. Phys. Rev. Lett. 56, 2415–2418 (1986).10.1103/PhysRevLett.56.2415 ↗
24. Sham, L. J. & Schlüter, M. Density-functional theory of the band gap. Phys. Rev. B 32, 3883–3889 (1985).10.1103/PhysRevB.32.3883 ↗
25. Sharp, R. T. & Horton, G. K. A variational approach to the unipotential many-electron problem. Phys. Rev. 90, 317–317 (1953).10.1103/PhysRev.90.317 ↗
26. Talman, J. D. & Shadwick, W. F. Optimized effective atomic central potential. Phys. Rev. A 14, 36–40 (1976).10.1103/PhysRevA.14.36 ↗
27. Städele, M., Moukara, M., Majewski, J., Vogl, P. & Görling, A. Exact exchange kohn-sham formalism applied to semiconductors. Phys. Rev. B 59, 10031 (1999).10.1103/PhysRevB.59.10031 ↗
28. Neumann, R., Nobes, R. H. & Handy, N. C. Exchange functionals and potentials. Mol. Phys. 87, 1–36 (1996).10.1080/00268979600100011 ↗
29. Paier, J. et al. Screened hybrid density functionals applied to solids. J. Chem. Phys. 124, 154709 (2006).10.1063/1.2187006 ↗
30. Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).10.1088/0953-8984/21/39/395502 ↗
31. Sun, J. et al. Self-consistent meta-generalized gradient approximation within the projector-augmented-wave method. Phys. Rev. B 84, 035117 (2011).10.1103/PhysRevB.84.035117 ↗
32. Yao, Y. & Kanai, Y. Plane-wave pseudopotential implementation and performance of scan meta-gga exchange-correlation functional for extended systems. J. Chem. Phys. 146, 224105 (2017).10.1063/1.4984939 ↗
33. Seidl, A., Görling, A., Vogl, P., Majewski, J. A. & Levy, M. Generalized Kohn–Sham schemes and the band-gap problem. Phys. Rev. B 53, 3764–3774 (1996).10.1103/PhysRevB.53.3764 ↗
34. Perdew, J. P. et al. Understanding band gaps of solids in generalized Kohn–Sham theory. PNAS 114, 2801–2806 (2017).10.1073/pnas.1621352114 ↗
35. Cohen, A. J., Mori-Sánchez, P. & Yang, W. Fractional charge perspective on the band gap in density-functional theory. Phys. Rev. B 77, 115123 (2008).10.1103/PhysRevB.77.115123 ↗
36. Mori-Sánchez, P., Cohen, A. J. & Yang, W. Localization and delocalization errors in density functional theory and implications for band-gap prediction. Phys. Rev. Lett. 100, 146401 (2008).10.1103/PhysRevLett.100.146401 ↗
37. Mori-Sánchez, P., Cohen, A. J. & Yang, W. Discontinuous nature of the exchange-correlation functional in strongly correlated systems. Phys. Rev. Lett. 102, 066403 (2009).10.1103/PhysRevLett.102.066403 ↗
38. Yang, W., Cohen, A. J. & Mori-Sánchez, P. Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory. J. Chem. Phys. 136, 204111 (2012).10.1063/1.3702391 ↗
39. Marques, M. A. L., Vidal, J., Oliveira, M. J. T., Reining, L. & Botti, S. Density-based mixing parameter for hybrid functionals. Phys. Rev. B 83, 035119 (2011).10.1103/PhysRevB.83.035119 ↗
40. Salpeter, E. E. & Bethe, H. A. A relativistic equation for bound-state problems. Phys. Rev. 84, 1232–1242 (1951).10.1103/PhysRev.84.1232 ↗
41. Runge, E. & Gross, E. K. U. Density-functional theory for time-dependent systems. Phys. Rev. Lett. 52, 997–1000 (1984).10.1103/PhysRevLett.52.997 ↗
42. Chen, W., Miceli, G., Rignanese, G.-M. & Pasquarello, A. Nonempirical dielectric-dependent hybrid functional with range separation for semiconductors and insulators. Phys. Rev. Mater. 2, 073803 (2018).10.1103/PhysRevMaterials.2.073803 ↗
43. Borlido, P. et al. Large-scale benchmark of exchange-correlation functionals for the determination of electronic band gaps of solids. J. Chem. Theory Comput. 15, 5069–5079 (2019).10.1021/acs.jctc.9b00322 ↗
44. Krukau, A. V., Vydrov, O. A., Izmaylov, A. F. & Scuseria, G. E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys. 125, 224106 (2006).10.1063/1.2404663 ↗
45. Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J. & Sutton, A. P. Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA+U study. Phys. Rev. B 57, 1505–1509 (1998).10.1103/PhysRevB.57.1505 ↗
46. Liechtenstein, A. I., Anisimov, V. I. & Zaanen, J. Density-functional theory and strong interactions: orbital ordering in Mott-Hubbard insulators. Phys. Rev. B 52, R5467–R5470 (1995).10.1103/PhysRevB.52.R5467 ↗
47. Doumont, J., Tran, F. & Blaha, P. Limitations of the DFT–1/2 method for covalent semiconductors and transition-metal oxides. Phys. Rev. B 99, 115101 (2019).10.1103/PhysRevB.99.115101 ↗
48. Ferreira, L. G., Marques, M. & Teles, L. K. Approximation to density functional theory for the calculation of band gaps of semiconductors. Phys. Rev. B 78, 125116 (2008).10.1103/PhysRevB.78.125116 ↗
49. Xue, K.-H., Yuan, J.-H., Fonseca, L. R. & Miao, X.-S. Improved LDA-1/2 method for band structure calculations in covalent semiconductors. Comput. Mater. Sci. 153, 493–505 (2018).10.1016/j.commatsci.2018.06.036 ↗
50. Colonna, N., Nguyen, N. L., Ferretti, A. & Marzari, N. Koopmans–Compliant functionals and potentials and their application to the GW100 test set. J. Chem. Theory Comput. 15, 1905–1914 (2019).10.1021/acs.jctc.8b00976 ↗
51. Das, T., Di Liberto, G., Tosoni, S. & Pacchioni, G. Band gap of 3D metal oxides and quasi-2D materials from hybrid density functional theory: are dielectric-dependent functionals superior? J. Chem. Theory Comput. 15, 6294–6312 (2019).10.1021/acs.jctc.9b00545 ↗
52. Liu, P., Franchini, C., Marsman, M. & Kresse, G. Assessing model-dielectric-dependent hybrid functionals on the antiferromagnetic transition-metal monoxides MnO, FeO, CoO, and NiO. J. Phys.: Condens. Matter 32, 015502 (2020).
53. Tsuneda, T. & Hirao, K. Self-interaction corrections in density functional theory. J. Chem. Phys. 140, 18A513 (2014).10.1063/1.4866996 ↗
54. Schmidt, J., Marques, M. R. G., Botti, S. & Marques, M. A. L. Recent advances and applications of machine learning in solid-state materials science. npj Comput. Mater. 5, 1–36 (2019).10.1038/s41524-019-0221-0 ↗
55. Xie, T. & Grossman, J. C. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Phys. Rev. Lett. 120, 145301 (2018).10.1103/PhysRevLett.120.145301 ↗
56. Isayev, O. et al. Universal fragment descriptors for predicting properties of inorganic crystals. Nat. Commun. 8, 15679 (2017).10.1038/ncomms15679 ↗
57. Rajan, A. C. et al. Machine-learning-assisted accurate band gap predictions of functionalized MXene. Chem. Mater. 30, 4031–4038 (2018).10.1021/acs.chemmater.8b00686 ↗
58. Zhuo, Y., Tehrani, A. M. & Brgoch, J. Predicting the band gaps of inorganic solids by machine learning. J. Phys. Chem. Lett. 9, 1668–1673 (2018).10.1021/acs.jpclett.8b00124 ↗
59. Perdew, J. P. & Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23, 5048–5079 (1981).10.1103/PhysRevB.23.5048 ↗
60. Perdew, J. P. et al. Restoring the density-gradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008).10.1103/PhysRevLett.100.136406 ↗
61. Becke, A. D. & Johnson, E. R. A simple effective potential for exchange. J. Chem. Phys. 124, 221101 (2006).10.1063/1.2213970 ↗
62. Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).10.1103/PhysRevLett.115.036402 ↗
63. Freysoldt, C. et al. First-principles calculations for point defects in solids. Rev. Mod. Phys. 86, 253–305 (2014).10.1103/RevModPhys.86.253 ↗
64. Adamo, C. & Barone, V. Toward reliable density functional methods without adjustable parameters: the PBE0 model. J. Chem. Phys. 110, 6158–6170 (1999).10.1063/1.478522 ↗
65. Perdew, J. P., Ernzerhof, M. & Burke, K. Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys. 105, 9982–9985 (1996).10.1063/1.472933 ↗
66. Perdew, J. P. & Schmidt, K. Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conf. Proc. 577, 1 (2001).10.1063/1.1390175 ↗
67. Finzel, K. & Baranov, A. I. A simple model for the Slater exchange potential and its performance for solids. Int. J. Quantum Chem. 117, 40–47 (2017).10.1002/qua.25312 ↗
68. Kohn, W. Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett. 76, 3168–3171 (1996).10.1103/PhysRevLett.76.3168 ↗
69. Slater, J. C., Wilson, T. M. & Wood, J. H. Comparison of several exchange potentials for electrons in the Cu+ ion. Phys. Rev. 179, 28–38 (1969).10.1103/PhysRev.179.28 ↗
70. Slater, J. C. A Simplification of the Hartree-Fock method. Phys. Rev. 81, 385–390 (1951).10.1103/PhysRev.81.385 ↗
71. Clementi, E. & Roetti, C. Roothaan-Hartree-Fock atomic wavefunctions. Data Nucl. Data Tables 14, 177–478 (1974).10.1016/S0092-640X(74)80016-1 ↗
72. Levy, M. & Perdew, J. P. 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 32, 2010–2021 (1985).10.1103/PhysRevA.32.2010 ↗
73. Engel, E. & Vosko, S. H. Exact exchange-only potentials and the virial relation as microscopic criteria for generalized gradient approximations. Phys. Rev. B 47, 13164–13174 (1993).10.1103/PhysRevB.47.13164 ↗
74. Ou-Yang, H. & Levy, M. Theorem for exact local exchange potential. Phys. Rev. Lett. 65, 1036–1039 (1990).10.1103/PhysRevLett.65.1036 ↗
75. Tran, F., Blaha, P. & Schwarz, K. Band gap calculations with Becke-Johnson exchange potential. J. Phys.: Condens. Matter 19, 196208 (2007).
76. Tran, F. & Blaha, P. Importance of the kinetic energy density for band gap calculations in solids with density functional theory. J. Phys. Chem. A 121, 3318–3325 (2017).10.1021/acs.jpca.7b02882 ↗
77. Tran, F. et al. Semilocal exchange-correlation potentials for solid-state calculations: current status and future directions. J. Appl. Phys. 126, 110902 (2019).10.1063/1.5118863 ↗
78. Armiento, R. & Mattsson, A. E. Functional designed to include surface effects in self-consistent density functional theory. Phys. Rev. B 72, 085108 (2005).10.1103/PhysRevB.72.085108 ↗
79. Kohn, W. & Mattsson, A. E. Edge electron gas. Phys. Rev. Lett. 81, 3487–3490 (1998).10.1103/PhysRevLett.81.3487 ↗
80. Mattsson, A. E. et al. The AM05 density functional applied to solids. J. Chem. Phys. 128, 084714 (2008).10.1063/1.2835596 ↗
81. Peverati, R., Zhao, Y. & Truhlar, D. G. Generalized gradient approximation that recovers the second-order density-gradient expansion with optimized across-the-board performance. J. Phys. Chem. Lett. 2, 1991–1997 (2011).10.1021/jz200616w ↗
82. Hammer, B., Hansen, L. B. & Nørskov, J. K. Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals. Phys. Rev. B 59, 7413–7421 (1999).10.1103/PhysRevB.59.7413 ↗
83. Perdew, J. P. In Proceedings of the WE-Heraeus-Seminar and 21st Annual International Symposium on Electronic Structure of Solids (eds Ziesche, P. & Eschrig, H.) 11 (Akademie Verlag, Berlin, 1991).
84. Perdew, J. P. et al. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 46, 6671–6687 (1992).10.1103/PhysRevB.46.6671 ↗
85. Perdew, J. P. et al. Erratum: Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 48, 4978–4978 (1993).10.1103/PhysRevB.48.4978.2 ↗
86. Peverati, R. & Truhlar, D. G. Performance of the M11-L density functional for bandgaps and lattice constants of unary and binary semiconductors. J. Chem. Phys. 136, 134704 (2012).10.1063/1.3698285 ↗
87. Armiento, R. & Kümmel, S. Orbital localization, charge transfer, and band gaps in semilocal density-functional theory. Phys. Rev. Lett. 111, 036402 (2013).10.1103/PhysRevLett.111.036402 ↗
88. Vlček, V., Steinle-Neumann, G., Leppert, L., Armiento, R. & Kümmel, S. Improved ground-state electronic structure and optical dielectric constants with a semilocal exchange functional. Phys. Rev. B 91, 035107 (2015).10.1103/PhysRevB.91.035107 ↗
89. Aschebrock, T., Armiento, R. & Kümmel, S. Orbital nodal surfaces: topological challenges for density functionals. Phys. Rev. B 95, 245118 (2017).10.1103/PhysRevB.95.245118 ↗
90. Aschebrock, T., Armiento, R. & Kümmel, S. Challenges for semilocal density functionals with asymptotically nonvanishing potentials. Phys. Rev. B 96, 075140 (2017).10.1103/PhysRevB.96.075140 ↗
91. Garhammer, J., Hofmann, F., Armiento, R. & Kümmel, S. On the challenge to improve the density response with unusual gradient approximations. Eur. Phys. J. B 91, 159 (2018).10.1140/epjb/e2018-90119-4 ↗
92. Perdew, J. P. & Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, 13244 (1992).10.1103/PhysRevB.45.13244 ↗
93. van Leeuwen, R. & Baerends, E. J. Exchange-correlation potential with correct asymptotic behavior. Phys. Rev. A 49, 2421–2431 (1994).10.1103/PhysRevA.49.2421 ↗
94. Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38, 3098–3100 (1988).10.1103/PhysRevA.38.3098 ↗
95. Gaiduk, A. P. & Staroverov, V. N. How to tell when a model Kohn-Sham potential is not a functional derivative. J. Chem. Phys. 131, 044107 (2009).10.1063/1.3176515 ↗
96. Karolewski, A., Armiento, R. & Kümmel, S. Electronic excitations and the Becke-Johnson potential: the need for and the problem of transforming model potentials to functional derivatives. Phys. Rev. A 88, 052519 (2013).10.1103/PhysRevA.88.052519 ↗
97. Cerqueira, T. F. T., Oliveira, M. J. T. & Marques, M. A. L. Benchmarking the AK13 exchange functional: ionization potentials and electron affinities. J. Chem. Theory Comput. 10, 5625–5629 (2014).10.1021/ct500550s ↗
98. Choi, J., Chang, E., Anstine, D. M., Madjet, M. E.-A. & Chakraborty, H. S. Effects of exchange-correlation potentials on the density-functional description of C 60 versus C 240 photoionization. Phys. Rev. A 95, 023404 (2017).10.1103/PhysRevA.95.023404 ↗
99. Marques, M. A. L., Castro, A. & Rubio, A. Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theory. J. Chem. Phys. 115, 3006–3014 (2001).10.1063/1.1385368 ↗
100. Perdew, J. P., Kurth, S., Zupan, A. & Blaha, P. Accurate density functional with correct formal properties: a step beyond the generalized gradient approximation. Phys. Rev. Lett. 82, 2544–2547 (1999).10.1103/PhysRevLett.82.2544 ↗