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The MC-DFT Approach and New Minnesota-type Functionals

This article discusses the MC-DFT approach, including the SCS-MP2 energies, to the DSD-PLYP method and new Minnesota-type functionals. It provides a historical background and explores various implementation of exchange-correlation functionals. It also introduces double hybrid DFT methods and the SCS-MP2 approach. Additionally, it discusses the development of Minnesota-type functionals and their performance in both chemistry and solid-state physics. The article concludes with the introduction of multi-coefficient DFT methods and their application in combination with SCS-MP2 and MC-SCS-MP2 methods.

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The MC-DFT Approach and New Minnesota-type Functionals

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  1. The MC-DFT Approach Including the SCS-MP2 Energies to the DSD-PLYP and New Minnesota-type Functionals Wei-Ping Hu (胡維平) Department of Chemistry and Biochemistry National Chung Cheng University (國立中正大學化學暨生物化學系) Jan. 13, 2014

  2. Historical BackgroundIs there an alternative to wavefunction? • 1964 Pierre Hohenberg and Walter Kohn:Density  Hamiltonian  Wavefunction • 1965 Kohn-Sham (KS) Method:noninteracting electrons experiencing an effective external potential  true density

  3. Historical Background

  4. Historical Background exchange,correlation &other corrections minimizing thetotal energy

  5. Jacob’s Ladder Heaven ? Various Implementation of the Exchange-Correlation Functional Triply Hybrid? EcMP4 Double hybrid ExHF, EcMP2 MC3BB, B2-PLYP, DSD-PLYP Hybrid ExHF B3LYP, B98 mGGA 2(r) TPSS, BB95 • GGA • r(r) BLYP • LSDA • (r) SVWN

  6. Exact (HF) Exchange and Hybrid DFT exact (HF) exchange energy: popular B3LYP functional:

  7. Doubly Hybrid DFT • 2004MC3-type theory proposed by TruhlarJournal of Physical Chemistry A 2004 108, 4786-4791 E(MC3BB) = c2[HF/6-31+G(d,p) + c1(MP2/6-31+G(d,p) - HF/6-31+G(d,p))] • + (1 -c2) (B1B95/MG3S) • E(MC3MPW) = c2[HF/6-31+G(d,p) + c1(MP2/6-31+G(d,p) - HF/6-31+G(d,p))] • + (1 -c2) (MPW1PW91/MG3S)c1 , c2, and X% for exact exchange were optimized

  8. Double Hybrid DFT • 2006Grimme proposed the first practical double hybrid GGA functional, namely, B2-PLYP • Exc = (1 -cx) Ex,B88+ cxEx,HF+ (1 - cc) Ec,LYP+ cc Ec,E2 • Ec,E2 is the MP2-like perturbational term based on the KS orbitals J. Chem. Phys.2006, 124, 034108. • 2006Instead of the B88, Grimme used mPW exchange and made a new double hybrid functional, namely, mPW2-PLYP • 2008Martin proposed the several reparametrizations such as B2K-PLYP, B2T-PLYP, B2GP-PLYPJ. Phys. Chem. A 2008, 112, 3; J. Phys. Chem. A2008, 112, 12868.

  9. SCS-MP2 • 2003Spin-component scaling MP2 (SCS-MP2) proposed by GrimmeJ. Chem. Phys.2003,118, 9095-9102 Ec,scs-MP2 = co Eoc,E2 + csEsc,E2 Esc : same spin correlation energy (αα, ββ, triplet or parallel-spin) Eoc : opposite spin correlation energy (αβ,singlet or antiparallel-spin) Optimized parameters co= 6/5 and cs= 1/3 Mean Unsigned Errors (kcal/mol) for the Various Methodsa a. cc-pVQZ AO basis b. The errors refer to the QCISD(T) value as reference

  10. DSD-BLYP • 2010DSD-BLYP functional proposed by Martin • J. Phys. Chem. C 2010, 114, 20801–20808 • Double hybrid + SCS-MP2 + Dispersion correction • Exc = (1 -cx) Ex,B88+cxEx,HF+cc Ec,LYP+ co Eoc,E2+csEsc,E2+ ED (sometimes omitted) RMSD (kcal/mol)

  11. Minnesota-type Functionals • Developed by the group of Prof. Donald Truhlar at the University of Minnesota. • 2006 Minnesota 06M06-L, M06, M06-2X and M06-HF (with 0%, 27%, 54% and 100% HF exchange, respectively) • 2008 Minnesota 08 • M08-HX (high percentages of nonlocal exact exchange) • M08-SO (second order gradient expansion) • 2011 Minnesota 11 • M11: Range-separated hybrid functional with 42.8% HF exchange in the short range and 100% in the long range. • 2012 Minnesota 12 • MN-12SX: Screened-exchange hybrid functional with 25% HF exchange in the short range and 0% in the long range.It provided good performance for both chemistry and solid-state physics.

  12. MUEs (kcal/mol-1) for the chemistry energetic databases a Multi-reference bond energies

  13. Multi-Coefficient DFT (MC-DFT) • 2008MC-DFTDensity Functional Methods with more than one basis sets (Journal of the Physical Chemistry A, 112, 1064-1070) • E2B = E(DFT/B1) + c1 [E(DFT/B2) – E(DFT/B1)] • E3B= E(DFT/B1) + c1[E(DFT/B2) – E(DFT/B1)] + c2 [E(DFT/B3) – E(DFT/B1)] • c1, c2, X% are coefficients obtained by optimizing the mean unsigned error against accurate energy values in the training set.

  14. Our Multi-Coefficient DFT with SCS-MP2 • E1B (SCS-MP2 | DFT) ＝ c1(HF/apdz＋co/apdz＋cs/apdz) ＋ (1－c1) E(DFT/B1) • E3B(SCS-MP2 | MC-DFT) ＝ c1 (HF/apdz＋co/apdz＋cs/apdz) ＋ (1－c1) {E(DFT/B1) ＋c2 [E(DFT/B2) －E(DFT/B1)] ＋c3 [E(DFT/B3) －E(DFT/B1)]} SCS-MP2/apdz

  15. Our Multi-Coefficient DFT with MC-SCS-MP2 • E1B (MC-SCS-MP2 | DFT) ＝ c1 {HF/pdz ＋co/pdz ＋cs/pdz ＋c2 (HF/apdz – HF/pdz) ＋c3 (HF/ptz – HF/pdz) ＋c4 [(co/apdz ＋cs/apdz)－ (co/pdz ＋cs/pdz)] ＋c5 [(co/ptz ＋cs/ptz) － (co/pdz ＋cs/pdz)]} ＋ (1－c1) E(DFT/B1) • E3B (MC-SCS-MP2 | MC-DFT) ＝ c1{HF/pdz ＋co/pdz ＋cs/pdz ＋c2 (HF/apdz – HF/pdz) ＋c3 (HF/ptz – HF/pdz) ＋c4 [(co/apdz ＋cs/apdz) － (co/pdz ＋cs/pdz)] ＋c5 [(co/ptz ＋cs/ptz) － (co/pdz ＋cs/pdz)]} ＋ (1－c1) {E(DFT/pdz) ＋c6 [E(DFT/apdz) －E(DFT/pdz)] ＋c7 [E(DFT/ptz) －E(DFT/pdz)]} SCS-MP2 (ptz/apdz/pdz)

  16. Basis Sets • Dunning-type basis sets cc-pVDZ(pdz) cc-pVTZ(ptz) aug-cc-pVDZ(apdz) aug-cc-pVTZ(aptz) • Pople-type basis sets 6-311+G(d,p)(TZdp) 6-311+G(2d,2p)(TZ2d2p) 6-311+G(2df,2pd)(TZ2df2pd) 6-311+G(3df,2pd)(TZ3df2pd) MG3S (G3large basis set without the core polarization functions and without the diffuse functions on hydrogens)

  17. Training Set • Thermochemical Kinetics Data 211 • 109 Main-Group Atomization Energies (MGAE109/11) • 38 Hydrogen Transfer Barrier Heights (HTBH 38/08) • 38 Non-Hydrogen Transfer Barrier Heights (NHTBH 38/08) • 13 Ionization Potentials (IP13/3) • 13 Electron Affinities (EA13/3)

  18. DSD-BLYP Mean Unsigned Errors (MUE) (kcal/mol) MP2 basis set DFT basis set

  19. MUE (kcal/mol) of MC-Minnesota Functionals

  20. MUE (kcal/mol) of SCS-MP2 | DFT and SCS-MP2 | MC-DFT method (MP2 basis set: apdz)

  21. MUE (kcal/mol) of MC-SCS-MP2 | DFT and MC-SCS-MP2 | MC-DFT method (MP2 basis set: pdz/apdz/ptz)

  22. MUEs (kcal/mol) of the B3LYP functional using various basis sets and basis set combinations. The tops of the green, red, and blue bars were the results by the MC-DFT, SCS-MP2 | MC-DFT, and MC-SCS-MP2 | MC-DFT approaches, respectively.

  23. MUE (kcal/mol), Relative Cost (%) and P/C Ratios of Several Efficient Methods using the B3LYP functional • Relative cost to M06-2X/aug-cc-pVTZ calculation. • Performance/cost (P/C) ratios were defined as 1 / (relative cost × MUE2).

  24. MUE (kcal/mol), Relative Cost (%) and P/C Ratios of Several Efficient Methods using the M06-2X functional

  25. MUE (kcal/mol), Relative Cost (%) and P/C Ratios of Several Efficient Methods using the M11 functional

  26. Acknowledgment Po-Chun Liu (劉柏均) Chia-Yu Peng(彭家瑜) Dr. Jien-Lian Chen(陳建良) Dr. Yi-Lun Sun(孫翊倫) Ji-Ting Hong(洪芷庭) Chun-Yu Yang(楊崇郁) Dr. Chun-Hui Li (李宗憓) Dr. Kuo-Jui Wu(吳國瑞) Chen-Chang Wu (吳承倉) NSC NCHC

  27. Thanks for your attention

  28. 中正大學

  29. 第六屆海峽兩岸理論化學研討會 2015

  30. 圖書資訊大樓參觀 研討會會場 (理學院二館) 學生活動中心 (晚餐) 致遠樓住宿

  31. 時間表

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