Minnesota Functionals (Myz) are a group of highly parameterized approximate exchange-correlation energy functionals in density functional theory (DFT). They are developed by the group of Prof. Donald Truhlar at the University of Minnesota. These functionals are based on the meta-GGA approximation, i.e. they include terms that depend on the kinetic energy density, and are all based on complicated functional forms parametrized on high-quality benchmark databases. These functionals can be used for traditional quantum chemistry and solid-state physics calculations. The Myz functionals are widely used and tested in the quantum chemistry community.[1][2][3][4]

Independent evaluations of the strengths and limitations of the Minnesota functionals with respect to various chemical properties have, however, cast doubts on the accuracy of Minnesota functionals.[5][6][7][8][9] Some regard this criticism to be unfair. In this view, because Minnesota functionals are aiming for a balanced description for both main-group and transition-metal chemistry, the studies assessing Minnesota functionals solely based on the performance on main-group databases[5][6][7][8] yield biased information, as the functionals that work well for main-group chemistry may fail for transition metal chemistry.

A study in 2017 highlighted the poor performance of Minnesota functionals on atomic densities.[10] Some others have refuted this criticism claiming that focusing only on atomic densities (including chemically unimportant, highly charged cations) is hardly relevant to real applications of density functional theory in computational chemistry. A recent study has found this to be the case: for Minnesota functionals (which are very popular in computational chemistry for calculating energy-related quantities), the errors in atomic densities and in energetics are indeed decoupled, and the Minnesota functionals perform better for diatomic densities than for the atomic densities.[11] The study[11] concludes that atomic densities do not yield an accurate judgement of the performance of density functionals. Minnesota functionals have also been shown to reproduce chemically relevant Fukui functions better than they do the atomic densities.[12]

Minnesota functionals are available in a large number of popular quantum chemistry computer programs.

Family of functionals

Minnesota 05

The first family of Minnesota functionals, published in 2005, is composed by:

  • M05:[13] Global hybrid functional with 28% HF exchange.
  • M05-2X[14] Global hybrid functional with 56% HF exchange.

In addition to the fraction of HF exchange, the M05 family of functionals includes 22 additional empirical parameters.[14] A range-separated functional based on the M05 form, ωM05-D which includes empirical atomic dispersion corrections, has been reported by Chai and coworkers.[15]

Minnesota 06

The '06 family represent a general improvement over the 05 family and is composed of:

  • M06-L:[16] Local functional, 0% HF exchange. Intended to be fast, good for transition metals, inorganic and organometallics.
  • revM06-L:[17] Local functional, 0% HF exchange. M06-L revised for smoother potential energy curves and improved overall accuracy.
  • M06:[18] Global hybrid functional with 27% HF exchange. Intended for main group thermochemistry and non-covalent interactions, transition metal thermochemistry and organometallics. It is usually the most versatile of the 06 functionals, and because of this large applicability it can be slightly worse than M06-2X for specific properties that require high percentage of HF exchange, such as thermochemistry and kinetics.
  • revM06:[19] Global hybrid functional with 40.4% HF exchange. Intended for a broad range of applications on main-group chemistry, transition-metal chemistry, and molecular structure prediction to replace M06 and M06-2X.
  • M06-2X:[18] Global hybrid functional with 54% HF exchange. It is the top performer within the 06 functionals for main group thermochemistry, kinetics and non-covalent interactions,[20] however it cannot be used for cases where multi-reference species are or might be involved,[20] such as transition metal thermochemistry and organometallics.
  • M06-HF:[21] Global hybrid functional with 100% HF exchange. Intended for charge transfer TD-DFT and systems where self-interaction is pathological.

The M06 and M06-2X functionals introduce 35 and 32 empirically optimized parameters, respectively, into the exchange-correlation functional.[18] A range-separated functional based on the M06 form, ωM06-D3 which includes empirical atomic dispersion corrections, has been reported by Chai and coworkers.[22]

Minnesota 08

The '08 family was created with the primary intent to improve the M06-2X functional form, retaining the performances for main group thermochemistry, kinetics and non-covalent interactions. This family is composed by two functionals with a high percentage of HF exchange, with performances similar to those of M06-2X:

  • M08-HX:[23] Global hybrid functional with 52.23% HF exchange. Intended for main group thermochemistry, kinetics and non-covalent interactions.
  • M08-SO:[23] Global hybrid functional with 56.79% HF exchange. Intended for main group thermochemistry, kinetics and non-covalent interactions.

Minnesota 11

The '11 family introduces range-separation in the Minnesota functionals and modifications in the functional form and in the training databases. These modifications also cut the number of functionals in a complete family from 4 (M06-L, M06, M06-2X and M06-HF) to just 2:

  • M11-L:[24] Local functional (0% HF exchange) with dual-range DFT exchange. Intended to be fast, to be good for transition metals, inorganic, organometallics and non-covalent interactions, and to improve much over M06-L.
  • M11:[25] Range-separated hybrid functional with 42.8% HF exchange in the short-range and 100% in the long-range. Intended for main group thermochemistry, kinetics and non-covalent interactions, with an intended performance comparable to that of M06-2X, and for TD-DFT applications, with an intended performance comparable to M06-HF.
  • revM11:[26] Range-separated hybrid functional with 22.5% HF exchange in the short-range and 100% in the long-range. Intended for good performance for electronic excitations and good predictions across the board for ground-state properties.

Minnesota 12

The 12 family uses a nonseparable[27] (N in MN) functional form aiming to provide balanced performance for both chemistry and solid-state physics applications. It is composed by:

  • MN12-L:[28] A local functional, 0% HF exchange. The aim of the functional was to be very versatile and provide good computational performance and accuracy for energetic and structural problems in both chemistry and solid-state physics.
  • MN12-SX:[29] Screened-exchange (SX) hybrid functional with 25% HF exchange in the short-range and 0% HF exchange in the long-range. MN12-L was intended to be very versatile and provide good performance for energetic and structural problems in both chemistry and solid-state physics, at a computational cost that is intermediate between local and global hybrid functionals.

Minnesota 15

The 15 family is the newest addition to the Minnesota family. Like the 12 family, the functionals are based on a non-separable form, but unlike the 11 or 12 families the hybrid functional doesn't use range separation: M15 is a global hybrid like in the pre-11 families. The 15 family consists of two functionals

  • MN15,[30] a global hybrid with 44% HF exchange.
  • MN15-L,[31] a local functional with 0% HF exchange.

Main Software with Implementation of the Minnesota Functionals

PackageM05M05-2XM06-LrevM06-LM06M06-2XM06-HFM08-HXM08-SOM11-LM11MN12-LMN12-SXMN15MN15-L
ADF Yes*Yes*YesNoYesYesYes Yes*Yes*Yes*Yes*Yes*Yes*Yes*Yes*
CPMD YesYesYesNoYesYesYes YesYesYesYesNoNoNoNo
GAMESS (US) YesYesYesNoYesYesYes YesYesYesYesYesYesYesYes
Gaussian 16 YesYesYesNoYesYesYes YesYesYesYesYesYesYesYes
Jaguar YesYesYesNoYesYesYes YesYesYesYesYesNoYesYes
Libxc YesYesYesYesYesYesYes YesYesYesYesYesYesYesYes
MOLCAS YesYesYesNoYesYesYes YesYesNoNoNoNoNoNo
MOLPRO YesYesYesNoYesYesYes YesYesYesNoNoNoNoNo
NWChem YesYesYesNoYesYesYes YesYesYesYesNoNoNoNo
Orca Yes*Yes*YesYes*YesYesYes* Yes*Yes*Yes*Yes*Yes*Yes*Yes*Yes*
PSI4 Yes*Yes*Yes*NoYes*Yes*Yes* Yes*Yes*Yes*Yes*Yes*Yes*Yes*Yes*
Q-Chem YesYesYesYesYesYesYes YesYesYesYesYesYesNoYes
Quantum ESPRESSO NoNoYesNoNoNoNo NoNoNoNoNoNoNoNo
TURBOMOLE Yes*Yes*YesYes*YesYesYes Yes*Yes*Yes*Yes*Yes*Yes*Yes*Yes*
VASP NoNoYesNoNoNoNo NoNoNoNoNoNoNoNo

* Using LibXC.

References

  1. ↑ A.J. Cohen, P. Mori-Sánchez and W. Yang (2012). "Challenges for Density Functional Theory". Chemical Reviews. 112 (1): 289–320. doi:10.1021/cr200107z. PMID 22191548.
  2. ↑ E.G. Hohenstein, S.T. Chill & C.D. Sherrill (2008). "Assessment of the Performance of the M05−2X and M06−2X Exchange-Correlation Functionals for Noncovalent Interactions in Biomolecules". Journal of Chemical Theory and Computation. 4 (12): 1996–2000. doi:10.1021/ct800308k. PMID 26620472.
  3. ↑ K.E. Riley; M Pitoňák; P. Jurečka; P. Hobza (2010). "Stabilization and Structure Calculations for Noncovalent Interactions in Extended Molecular Systems Based on Wave Function and Density Functional Theories". Chemical Reviews. 110 (9): 5023–63. doi:10.1021/cr1000173. PMID 20486691.
  4. ↑ L. Ferrighi; Y. Pan; H. Grönbeck; B. Hammer (2012). "Study of Alkylthiolate Self-assembled Monolayers on Au(111) Using a Semilocal meta-GGA Density Functional". Journal of Physical Chemistry. 116 (13): 7374–7379. doi:10.1021/jp210869r.
  5. 1 2 N. Mardirossian; M. Head-Gordon (2013). "Characterizing and Understanding the Remarkably Slow Basis Set Convergence of Several Minnesota Density Functionals for Intermolecular Interaction Energies". Journal of Chemical Theory and Computation. 9 (10): 4453–4461. doi:10.1021/ct400660j. OSTI 1407198. PMID 26589163. S2CID 206908565.
  6. 1 2 L. Goerigk (2015). "Treating London-Dispersion Effects with the Latest Minnesota Density Functionals: Problems and Possible Solutions". Journal of Physical Chemistry Letters. 6 (19): 3891–3896. doi:10.1021/acs.jpclett.5b01591. hdl:11343/209007. PMID 26722889.
  7. 1 2 N. Mardirossian; M. Head-Gordon (2016). "How accurate are the Minnesota density functionals for non-covalent interactions, isomerization energies, thermochemistry, and barrier heights involving molecules composed of main-group elements?". Journal of Chemical Theory and Computation. 12 (9): 4303–4325. doi:10.1021/acs.jctc.6b00637. OSTI 1377487. PMID 27537680. S2CID 5479661.
  8. 1 2 Taylor, DeCarlos E.; Ángyán, János G.; Galli, Giulia; Zhang, Cui; Gygi, Francois; Hirao, Kimihiko; Song, Jong Won; Rahul, Kar; Anatole von Lilienfeld, O. (2016-09-23). "Blind test of density-functional-based methods on intermolecular interaction energies". The Journal of Chemical Physics. 145 (12): 124105. Bibcode:2016JChPh.145l4105T. doi:10.1063/1.4961095. hdl:1911/94780. ISSN 0021-9606. PMID 27782652.
  9. ↑ Kepp, Kasper P. (2017-03-09). "Benchmarking Density Functionals for Chemical Bonds of Gold" (PDF). The Journal of Physical Chemistry A. 121 (9): 2022–2034. Bibcode:2017JPCA..121.2022K. doi:10.1021/acs.jpca.6b12086. ISSN 1089-5639. PMID 28211697. S2CID 206643889.
  10. ↑ Medvedev, Michael G.; Bushmarinov, Ivan S.; Sun, Jianwei; Perdew, John P.; Lyssenko, Konstantin A. (2017-01-06). "Density functional theory is straying from the path toward the exact functional". Science. 355 (6320): 49–52. Bibcode:2017Sci...355...49M. doi:10.1126/science.aah5975. ISSN 0036-8075. PMID 28059761. S2CID 206652408.
  11. 1 2 Brorsen, Kurt R.; Yang, Yang; Pak, Michael V.; Hammes-Schiffer, Sharon (2017). "Is the Accuracy of Density Functional Theory for Atomization Energies and Densities in Bonding Regions Correlated?". J. Phys. Chem. Lett. 8 (9): 2076–2081. doi:10.1021/acs.jpclett.7b00774. PMID 28421759.
  12. ↑ Gould, Tim (2017). "What Makes a Density Functional Approximation Good? Insights from the Left Fukui Function". J. Chem. Theory Comput. 13 (6): 2373–2377. doi:10.1021/acs.jctc.7b00231. hdl:10072/348655. PMID 28493684.
  13. ↑ Y. Zhao, N.E. Schultz & D.G. Truhlar (2005). "Exchange-correlation functional with broad accuracy for metallic and nonmetallic compounds, kinetics, and noncovalent interactions". Journal of Chemical Physics. 123 (16): 161103. Bibcode:2005JChPh.123p1103Z. doi:10.1063/1.2126975. PMID 16268672.
  14. 1 2 Y. Zhao, N.E. Schultz & D.G. Truhlar (2006). "Design of Density Functionals by Combining the Method of Constraint Satisfaction with Parametrization for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions". Journal of Chemical Theory and Computation. 2 (2): 364–382. doi:10.1021/ct0502763. PMID 26626525.
  15. ↑ Lin, You-Sheng; Tsai, Chen-Wei; Li, Guan-De & Chai, Jeng-Da (2012). "Long-range corrected hybrid meta-generalized-gradient approximations with dispersion corrections". Journal of Chemical Physics. 136 (15): 154109. arXiv:1201.1715. Bibcode:2012JChPh.136o4109L. doi:10.1063/1.4704370. PMID 22519317. S2CID 16662593.
  16. ↑ Y. Zhao & D.G. Truhlar (2006). "A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions". Journal of Chemical Physics. 125 (19): 194101. Bibcode:2006JChPh.125s4101Z. CiteSeerX 10.1.1.186.6548. doi:10.1063/1.2370993. PMID 17129083.
  17. ↑ Ying Wang; Xinsheng Jin; Haoyu S. Yu; Donald G. Truhlar & Xiao Hea (2017). "Revised M06-L functional for improved accuracy on chemical reaction barrier heights, noncovalent interactions, and solid-state physics". Proc. Natl. Acad. Sci. U.S.A. 114 (32): 8487–8492. Bibcode:2017PNAS..114.8487W. doi:10.1073/pnas.1705670114. PMC 5559035. PMID 28739954.
  18. 1 2 3 Y. Zhao & D.G. Truhlar (2008). "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. 120 (1–3): 215–241. doi:10.1007/s00214-007-0310-x.
  19. ↑ Y. Wang; P. Verma; X. Jin; D. G. Truhlar & X. He (2018). "Revised M06 density functional for main-group and transition-metal chemistry". Proc. Natl. Acad. Sci. U.S.A. 115 (41): 10257–10262. Bibcode:2018PNAS..11510257W. doi:10.1073/pnas.1810421115. PMC 6187147. PMID 30237285.
  20. 1 2 Mardirossian, Narbe; Head-Gordon, Martin (2017-10-02). "Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals". Molecular Physics. 115 (19): 2315–2372. Bibcode:2017MolPh.115.2315M. doi:10.1080/00268976.2017.1333644. ISSN 0026-8976.
  21. ↑ Y. Zhao & D.G. Truhlar (2006). "Density Functional for Spectroscopy: No Long-Range Self-Interaction Error, Good Performance for Rydberg and Charge-Transfer States, and Better Performance on Average than B3LYP for Ground States". Journal of Physical Chemistry A. 110 (49): 13126–13130. Bibcode:2006JPCA..11013126Z. doi:10.1021/jp066479k. PMID 17149824.
  22. ↑ Lin, You-Sheng; Li, Guan-De; Mao, Shan-Ping & Chai, Jeng-Da (2013). "Long-Range Corrected Hybrid Density Functionals with Improved Dispersion Corrections". J. Chem. Theory Comput. 9 (1): 263–272. arXiv:1211.0387. doi:10.1021/ct300715s. PMID 26589028. S2CID 13494471.
  23. 1 2 Y. Zhao & D.G. Truhlar (2008). "Exploring the Limit of Accuracy of the Global Hybrid Meta Density Functional for Main-Group Thermochemistry, Kinetics, and Noncovalent Interactions". Journal of Chemical Theory and Computation. 4 (11): 1849–1868. doi:10.1021/ct800246v. PMID 26620329.
  24. ↑ R. Peverati & D.G. Truhlar (2012). "M11-L: A Local Density Functional That Provides Improved Accuracy for Electronic Structure Calculations in Chemistry and Physics". Journal of Physical Chemistry Letters. 3 (1): 117–124. doi:10.1021/jz201525m.
  25. ↑ R. Peverati & D.G. Truhlar (2011). "Improving the Accuracy of Hybrid Meta-GGA Density Functionals by Range Separation". Journal of Physical Chemistry Letters. 2 (21): 2810–2817. doi:10.1021/jz201170d.
  26. ↑ P. Verma; Y. Wang; S. Ghosh; X. He & D. G. Truhlar (2019). "Revised M11 Exchange-Correlation Functional for Electronic Excitation Energies and Ground-State Properties". Journal of Physical Chemistry A. 123 (13): 2966–2990. Bibcode:2019JPCA..123.2966V. doi:10.1021/acs.jpca.8b11499. PMID 30707029. S2CID 73431138.
  27. ↑ R. Peverati & D.G. Truhlar (2012). "Exchange–Correlation Functional with Good Accuracy for Both Structural and Energetic Properties while Depending Only on the Density and Its Gradient". Journal of Chemical Theory and Computation. 8 (7): 2310–2319. doi:10.1021/ct3002656. PMID 26588964.
  28. ↑ R. Peverati & D.G. Truhlar (2012). "An improved and broadly accurate local approximation to the exchange–correlation density functional: The MN12-L functional for electronic structure calculations in chemistry and physics". Physical Chemistry Chemical Physics. 14 (38): 13171–13174. Bibcode:2012PCCP...1413171P. doi:10.1039/c2cp42025b. PMID 22910998.
  29. ↑ R. Peverati & D.G. Truhlar (2012). "Screened-exchange density functionals with broad accuracy for chemistry and solid-state physics". Physical Chemistry Chemical Physics. 14 (47): 16187–91. Bibcode:2012PCCP...1416187P. doi:10.1039/c2cp42576a. PMID 23132141.
  30. ↑ Yu, Haoyu S.; He, Xiao; Li, Shaohong L. & Truhlar, Donald G. (2016). "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. 7 (8): 5032–5051. doi:10.1039/C6SC00705H. PMC 6018516. PMID 30155154.
  31. ↑ Yu, Haoyu S.; He, Xiao & Truhlar, Donald G. (2016). "MN15-L: A New Local Exchange-Correlation Functional for Kohn–Sham Density Functional Theory with Broad Accuracy for Atoms, Molecules, and Solids". J. Chem. Theory Comput. 12 (3): 1280–1293. doi:10.1021/acs.jctc.5b01082. PMID 26722866.
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