Improvement of Multi-Coefficient Density Functional Theory with SCS-MP2 and E3-E4 Correlation Energy

1. 國立中正大學 化學暨生物化學研究所 博士班資格考試 第一階段口試 彭家瑜 (Chia-Yu Peng) 指導教授：胡維平 (Wei-Ping Hu) • 中華民國 102 年 7 月 29 日

2. Abstract In this study, we improved our multi-coefficient density functional theory (MC-DFT) by combining the spin-component-scaled second-order Møller-Plesset perturbation (SCS-MP2) energy, and further improved it by combining the E3 and E4 correlation energy with our method. We tested our methods on the DSD-BLYP functional which had the best accuracy in our MC-DFT. In our test, the DSD-BLYP calculated with cc-pVDZ, cc-pVTZ and aug-cc-pVDZ basis set combination and corrected by SCS-MP2/aug-cc-pVDZ energy can achieve accuracy of mean unsigned error (MUE) 1.19 kcal/mol for the TK211 set. When the SCS-MP2 calculated with the same basis set combination as mentioned above and corrected by MP4 energy, an astonishingly small MUE of 0.92 and 0.73 kcal/mol were achieved. The computational costs of the new methods were only 60% and 78% of the cost of DSD-BLYP/aug-cc-pVTZ but accuracy was improved significantly (by 1.47 kcal/mol).

3. Jacob’s Ladder Heaven Various Implementation of the Exchange-Correlation Functionalfor Density Functional Theory (DFT) Double hybrid ExHF, EcE2 MC3BB, B2-PLYP, DSD-BLYP Hybrid ExHF B3LYP, B98 mGGA 2(r) TPSS, BB95, M06-L • GGA • r(r) BLYP • LSDA • (r) SVWN

4. Hybrid DFT exact (HF) exchange energy:

5. Doubly-Hybrid DFT • 2004MC3-type theory proposed by TruhlarJ. Phys. Chem. A 2004, 108, 4786. E(MC3BB) = c2[HF/DIDZ + c1(MP2/DIDZ - HF/DIDZ)] • + (1 -c2) (B1B95/MG3S) • E(MC3MPW) = c2[HF/DIDZ + c1(MP2/DIDZ - HF/DIDZ)] • + (1 -c2) (MPW1PW91/MG3S)c1 , c2, and X% for exact exchange were optimized. • DIDZ represents 6-31+G(d,p).

6. Doubly-Hybrid DFT Mean Errors for Barriers (kcal/mol) and Atomization Energies per Bond (kcal/mol)

7. Double-Hybrid DFT (DHDFT) • 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 orbitalsJ. Chem. Phys.2006, 124, 034108. • 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. Multi-Coefficient DFT (MC-DFT) • 1998G3 methodMultilevel Methods • 1999 MCG3Multilevel Methods with Scaled Energies • 2005MCG3-DFT • 2008 MC-DFTDensity Functional Methods with more than one basis sets • 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.

10. Mean Unsigned Errors (MUE, kcal/mol) of the MC-DFT Methods for the Training Set (TK211) *The pdz/ptz/apdz and pc1/pc2/apc1 basis set combinationsnot onlyreduce the computational cost but improve the accuracy. *The pdz/MG3S with almost the same cost but improve the accuracy. J. Phys. Chem. A 2008, 112, 1064.

11. MUE (kcal/mol) of the MC-DFT Methods for TK211 *The pdz/ptz/apdz and pc1/pc2/apc1 basis set combinationsnot onlyreduce the computational cost but improve the accuracy. *The pdz/MG3S with almost the same cost but improve the accuracy. Chemical Physics Letters 2009, 468, 307.

12. MC-DFT: Basis Set Extrapolation to an Optimal Size at an Affordable Cost Cost  N4 Accuracy MC-DFT Basis Set Size (N)of DFT

13. SCS-MP2 • 2003Spin-component scaling MP2 (SCS-MP2) proposed by GrimmeJ. Chem. Phys.2003,118, 9095. • Ec,scs-E2 = co Eoc,E2 + csEsc,E2 Esc : same spin correlation energy (αα, ββ, or parallel-spin) Eoc : opposite spin correlation energy (αβ,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

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

15. Current Work(Triply-Hybrid DFT) • MC-DFT with SCS-MP2 Energy corrections for multi-coefficient B1B95, MPW1B95, MPW1PW91, TPSS1KCIS, B98, B3LYP, M06-2X and DSD-BLYP (2) MC-DSD-BLYP with MC-SCS-MP2 Energy corrections for multi-coefficient DSD-BLYP (3) MC-DSD-BLYP with MC-MP4 (MC-MP4 includedtheSCS-MP2 energy) Energy corrections for multi-coefficient DSD-BLYP

16. Training/Test Sets • Thermochemical Kinetics Data 211 (TK211) • 109 Main-Group Atomization Energies (MGAE109/05) • 38 Hydrogen Transfer Barrier Heights (HTBH 38/04) • 38 Non-Hydrogen Transfer Barrier Heights (NHTBH 38/04) • 13 Ionization Potentials (IP13/3) • 13 Electron Affinities (EA13/3)

17. Training Sets MGAE109/05 database of zero-point-exclusive atomization energies (kcal/mol)

18. HTBH38/04 database (kcal/mol) NHTBH38/04 databases (kcal/mol) a.Vf≠ denote forward BH and Vr≠ denote reverse BH

19. Zero-point-exclusive ionization potentials (IP13/3) and electron affinities (EA13/3) databases (kcal/mol)

20. Basis Sets in MC-DFT • Dunning-type correlation-consistent basis setscc-pVDZ (pdz) cc-pVTZ (ptz)aug-cc-pVDZ (apdz)aug-cc-pVTZ(aptz) • Pople-type basis setsMG3S • Jensen’s basis sets (designed for HF, DFT)pc1、pc2、aug-pc1(apc1)、apc2、apc3

21. Basis Sets in MP2 Correction Energies MUE (kcal/mol) of the MC-SCS-MP2|MC-DSD-BLYP Methods MPn basis DFT basis MPn basis DFT basis *The apdz and pdz/ptz/apdz basis set (combination) provide the best accuracy and affordable cost in the MP2 calculations.

22. (1) MC-DFT with SCS-MP2 E(MP2 | MC-DFT) = c2(HF/apdz + c1E2/apdz) + (1 -c2) (MC-DFT)E(SCS-MP2 | MC-DFT) = c2(HF/apdz + co E2o/apdz +csE2s/apdz) + (1 -c2) (MC-DFT) SCS-E2/apdz The SCS-MP2 calculation was carried out with aug-cc-pVDZ basis set.

23. (2) MC-DSD-BLYP with MC-SCS-MP2 E(MC-SCS-MP2 | MC-DSD-BLYP) = c6{HF/pdz + c1(SCS-E2/pdz) + c2(HF/apdz – HF/pdz) + c3(HF/ptz – HF/pdz) + c4[SCS-E2/apdz – SCS-E2/pdz] + c5[SCS-E2/ptz – SCS-E2/pdz]} + (1 – c6) (MC-DSD-BLYP) The MC-SCS-MP2 calculation was carried out with cc-pVDZ, cc-pVTZ and aug-cc-pVDZ basis set combination.

24. (3) MC-DSD-BLYP with MC-MP4 E(MC-MP4 | MC-DSD-BLYP) = c12{HF/pdz + c1(SCS-E2/pdz) + c2E3/pdz + c3E4/pdz + c4(HF/apdz – HF/pdz) + c5(HF/ptz – HF/pdz) + c6[SCS-E2/apdz – SCS-E2/pdz] + c7[SCS-E2/ptz – SCS-E2/pdz] + c8(E3/apdz – E3/pdz) + c9(E3/ptz – E3/pdz) + c10(E4/apdz – E4/pdz) + c11(E4/ptz – E4/pdz)} + (1 -c12) (MC-DSD-BLYP) The MC-MP4 calculation was carried out with cc-pVDZ, cc-pVTZ and aug-cc-pVDZ basis set combination.

25. Preliminary Results MUE (kcal/mol) of the SCS-MP2/apdz|MC-DFT Methods *The basis set combinationand MP2 correctionsignificantlyimprove the accuracy for these four functionals.

26. MUE (kcal/mol) of the SCS-MP2/apdz|MC-DFT Methods *First DFT-based method with triple-z basis set to reach MUE < 1.0 kcal/mol for Thermochemical Kinetics Benchmark

27. MUE (kcal/mol) of the MC-SCS-MP2|MC-DSD-BLYP Methods MPn basis DFT basis DFT = DSD-PLYP without dispersion *MC-DSD-BLYP calculated with dunning-type basis set (combinations) and apc3 including MC-SCS-MP2 correction achieve the MUE of ~0.9 kcal/mol.

28. Computational Cost (s) and MUE (kcal/mol)of theMC-SCS-MP2|MC-DSD-BLYP Methods MPn basis DFT basis *MC-DSD-BLYP calculated with pdz/ptz/apdz basis set combination provide the best efficiency in MC-SCS-MP2|MC-DSD-BLYP methods.

29. MUE (kcal/mol) of the MPn|MC-DSD-BLYP Methods MPn basis DFT basis *MC-DSD-BLYP calculated with dunning-type basis set (combinations) and apc3 including MC-MP4 correction achieve the MUE of ~0.7 kcal/mol.

30. Computational Cost (s) and MUE (kcal/mol)of the MC-MP4|MC-DSD-BLYP Methods MPn basis DFT basis *MC-DSD-BLYP calculated with pdz/ptz/apdz basis set combination provide the best efficiency in MC-MP4|MC-DSD-BLYP methods.

31. MUE(kcal/mol)and Relative Cost (%) of the Several Efficient Methods using Various Basis Sets (Combinations) in MC-DSD-BLYP MUE (kcal/mol) 1.00 1.50 0.00 Relative Cost (%)

32. Conclusions • We have developed some efficient methods combining the MC-DHDFT with MPn energies called “triply-hybrid DFT”. • The SCS-MP2 | MC-DSD-BLYP (pdz/ptz/apdz) provides the lowest cost (39% of cost relative to DSD-BLYP/aptz) in this study and 1.19 kcal/mol of MUE on TK211 set. • The MC-MP4 | MC-DSD-BLYP (pdz/ptz/apdz) provides the best accuracy (0.72 kcal/mol of MUE on TK211 set) in this study and 78% of cost relative to DSD-BLYP/aptz.

33. Future Works • We will try to improve our methods by E3 and E4 correction energies. • We will use the bigger systems for training/test set including the transition metal, van der Waals, hydrogen-bonds and weak interaction complexes. • We will simplify the basis set like aug-cc-pVTZ in MLSE(C2)-DFT to reduce the computational cost of our methods.

34. Acknowledgment • Prof. Wei-Ping Hu • Prof. Kuo-Jui Wu • Our group members. (Dr. Jien-Lian Chen, Dr. Yi-Lun Sun et al.) • Department of Chemistry & Biochemistry, National Chung Cheng University • National Center for High-Performance Computing

35. Thanks for your attention

36. Density Functional Theory (DFT) • 1964Pierre Hohenberg and Walter Kohn Density  Hamiltonian  Wavefunction • 1965Kohn-Sham (KS) Methodnoninteracting electrons experiencing an effective external potential  true density

37. Density Functional Theory (DFT)

38. Density Functional Theory (DFT) exchange,correlation &other corrections minimizing thetotal energy