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Advanced Density Functional Theory Methods for Chemical Kinetics Study

This study focuses on enhancing multi-coefficient density functional theory (MC-DFT) by incorporating energies from second-order Møller-Plesset perturbation theory (SCS-MP2) and E3/E4 correlation energies. We evaluated various functionals using different basis sets and corrections, achieving impressive accuracy for chemical kinetics calculations. The application of our method with the DSD-BLYP functional yielded exceptional results. We discuss the development and optimization of double hybrid functionals and highlight the importance of MC-DFT in improving accuracy and reducing computational costs. Various MC-DFT methods and their applications are detailed, including basis set extrapolation and the incorporation of SCS-MP2 corrections. Our work explores new avenues in the field, showcasing the potential of triply-hybrid DFT for advanced chemical kinetics studies.

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Advanced Density Functional Theory Methods for Chemical Kinetics Study

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  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 energies with our method. In our test, the B1B95, MPW1B95 and M06-2X functionals 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 errors below 1.3 kcal/mol for the TK211 set. When the new method (MC-DFT corrected by SCS-MP2/aug-cc-pVDZ) was used with the new double hybrid functional DSD-BLYP, an astonishingly small mean unsigned errors of 0.98 kcal/mol was achieved using the cc-pVDZ, aug-cc-pVDZ and aug-cc-pVTZ basis set combination.

  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.

  8. 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.

  9. 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.

  10. 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 has almost the same cost but improve the accuracy. Chemical Physics Letters 2009, 468, 307.

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

  12. 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

  13. 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)

  14. 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-DFT with MC-SCS-MP2 (3) MC-DFT with MC-MP4 (MC-MP4 includedtheSCS-MP2 energy)

  15. 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)

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

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

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

  19. 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

  20. Basis Sets in MP2 Correction Energies MUE (kcal/mol) of the SCS-MP2|MC-DSD-BLYP and 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.

  21. (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.

  22. (2) MC-DFT with MC-SCS-MP2 E(MC-SCS-MP2 | MC-DFT) = 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-DFT) The MC-SCS-MP2 calculation was carried out with cc-pVDZ, cc-pVTZ and aug-cc-pVDZ basis set combination.

  23. (3) MC-DFT with MC-MP4 E(MC-MP4 | MC-DFT) = 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-DFT) The MC-MP4 calculation was carried out with cc-pVDZ, cc-pVTZ and aug-cc-pVDZ basis set combination.

  24. Preliminary Results MUE (kcal/mol) of the SCS-MP2/apdz|MC-DFT Methods *MC-DFT calculated with dunning-type basis set combinations including SCS-MP2 correction achieve the MUE ~1.6 kcal/mol.

  25. MUE (kcal/mol) of the SCS-MP2/apdz|MC-DFT Methods *MC-DFT calculated with dunning-type basis set combinations including SCS-MP2 correction achieve the MUE < 1.3 kcal/mol.

  26. MUE (kcal/mol) of the SCS-MP2/apdz|MC-B3LYP Methods

  27. MUE (kcal/mol) of the SCS-MP2/apdz|MC-B98 Methods

  28. MUE (kcal/mol) of the SCS-MP2/apdz|MC-BMK Methods

  29. Computational Cost (s) and MUE (kcal/mol)of theSCS-MP2|MC-M06-2X Methods

  30. Computational Cost (s) and MUE (kcal/mol)of theSCS-MP2|MC-DSD-BLYP Methods *First DFT-based method with triple-z basis set to reach MUE < 1.0 kcal/mol for Thermochemical Kinetics Benchmark

  31. Conclusions • We have developed some efficient methods combining the MC-DHDFT with MP2 energies called “triply-hybrid DFT”. • The SCS-MP2 | MC-DFT with pdz/apdz/aptz basis set combination has been shown to be able to predict very accurate molecular energies at almost the same costs with DFT/aptz. In these methods for B1B95, MPW1B95 and M06-2X can achieve accuracies of MUE below 1.3 kcal/molon TK211 set, and the method for DSD-BLYP functional provides the best accuracy with a MUE of 0.98 kcal/molin this study on TK211 set.

  32. Conclusions • The SCS-MP2 | MC-DFT with pdz/ptz/apdz basis set combination has been shown to be able to improve the accuracies of DFT/aptz at 40% of cost of DFT/aptz. In this method for DSD-BLYP functional can achieve the MUE of 1.19 kcal/molon TK211 set.

  33. Future Works • We will try to improve our methods by E3 and E4 correction energies (MC-MP4 | MC-DFT) and further improve it by QCISD(T) and CCSD(T) correction energies. • We will use the bigger systems for training sets and test sets including 4th and 5th period elements such as the transition metals and heavy halogens. • We will simplify the Aug-cc-pVTZ basis set in our methods to reduce the computational cost of our methods. For example, we can use the Jun-cc-pVTZ which removes the highest angular momentum diffuse function from Aug-cc-pVTZ for all atoms in gaussian 09.

  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

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