Multicoefficient Density Functional Theory WeiPing Hu Department of

Multicoefficient Density Functional Theory Wei-Ping Hu (胡維平) Department of Chemistry and Biochemistry National Chung Cheng University Chia-Yi, Taiwan 1

History & Background • 1964 Pierre Hohenberg and Walter Kohn: Density Hamiltonian wavefunction • 1965 Kohn-Sham (KS) Method: noninteracting electrons experiencing an effective external potential 2

History & Background 3

History & Background 4

History & Background exact (HF) exchange energy: 5

Jacob’s Ladder Heaven ? (PT 2, virtual orbitals) Hybrid Ex. HF B 3 LYP, B 98 m. GGA 2 (r) TPSS, BB 95 GGA r(r) BLYP LSDA r(r) SVWN 6

Basis-Set Dependence from Curtiss, et. al. , J. Chem. Phys. 123, 124107 (2005) 7

Electronic Structure Calculation Electron correlation → Basis set Type HF MP 2 MP 3 MP 4 QCISD(T) … Full CI Minimal 　 　 　 Split-valence 　 　 　 Polarized 　 　 　 Diffuse 　 　 　 High ang. momentum … Complete Basis set … HF Limit 　 … 　 … 　 … Schrödinger Equation　

Multilevel Methods Base Calculation + Corrections for Incomplete Basis Set Incomplete Electron Correlation Currently Used Multilevel Methods: G 2, G 3, G 4, CBS HF, MP 2, MP 4, QCISD(T), empirical HLC 6 -31 G(d), 6 -311 G(d, p), 6 -311+G(d, p) 6 -311+G(2 df, p), 6 -311+G(3 df, 2 p), G 3 Large Multilevel Methods with Scaled Energies: (Multicoefficient Method) MCG 3, G 3 S, G 3 X 9

The G 3 Method • E(G 3) = E[QCISD(T)/6 -31 G(d)] + E(+) + E(2 df) + E(G 3 Large) + E(SO) + E(HLC) Journal of Chemical Physics, 109, 7764 (1998) 10

The MCG 3 Method • E(MCG 3/3) = c 0 E(HF/6 -31 G(d)) + c 1 E(HF/MG 3 S | 6 -31 G(d)) + c 2 E(MP 2 | HF/6 -31 G(d)) + c 3 E(MP 2 | HF/MG 3 S | 6 -31 G(d)) + c 4 E(MP 4 SDQ | MP 2/6 -31 G(d)) + c 5 E(MP 4 SDQ | MP 2/6 -31 G(2 df, p) | 6 -31 G(d)) + c 6 E(QCISD(T) | MP 4 SDQ/6 -31 G(d)) + ESO J. Phys. Chem. A 2003, 107, 3898. 11

The MLSE-DFT Method E(MLSE-DFT) = CWF { E(HF/cc-p. V(D+d)Z) + C HF [E(HF/cc-p. V(T+d)Z )– E(HF/cc-p. V(D+d)Z)] + CE 2 [E 2/cc-p. V(D+d)Z] + CE 34 [E(MP 4 SDQ/cc-p. V(D+d)Z) – E(MP 2/cc-p. V(D+d)Z)] + CQCI [E(QCISD(T)/cc-p. V(D+d)Z) – E(MP 4 SDQ/cc-p. V(D+d)Z)] + CB [E 2/cc-p. V(T+d)Z – E 2/cc-p. V(D+d)Z] + CHF+ [E(HF/aug-cc-p. V(D+d)Z) – E(HF/cc-p. V(D+d)Z]) + CE 2+ [E 2/aug-cc-p. V(D+d)Z – E 2/cc-p. V(D+d)Z] } + (1 - CWF ) { E(DFTX/cc-p. V(D+d)Z) + CB 1 [E(DFTX/cc-p. V(T+d)Z – DFTX/cc-p. V(D+d)Z] } + ESO Chem. Phys. Lett. 2007, 442, 220. 12

Accuracy 13

MLSE-DFT Optimized Coefficients 14

MLSE-DFT Computational Cost 15

Multi-Coefficient DFT • Density Functional Methods with more than one basis sets (without WFT) • E 2 B = E(DFT/B 1) + c 1 [ E(DFT/B 2) – E(DFT/B 1) ] • E 3 B = E(DFT/B 1) + c 1 [E(DFT/B 2) – E(DFT/B 1)]+ c 2 [E(DFT/B 3) – E(DFT/B 1)] • c 1 , c 2 , and the mixing percentage (X%) of the exact exchange energies are determined by minimizing the mean unsigned errors (MUEs) relative to the database values (the training sets) 16

Functionals and Basis Sets • Tested Functionals: B 1 B 95, MPW 1 PW 91, TPSS 1 KCIS, B 98, B 3 LYP, BMK • Tested Basis-Set Combination cc-p. VDZ/cc-p. VTZ/aug-cc-p. VDZ 6 -31+G**/MG 3 S cc-p. VDZ/MG 3 S 17

MC-DFT Training Sets 18

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MC-DFT Extrapolation Accuracy Cost N 4 MC-DFT Basis Set Size N 27

Future Work • MC-Double-Hybrid-DFT • MC-Doubly-Hybrid-DFT • Dual-Functional MC-DFT • Semiempirical Hybrid-DFT Optimized for MC -DFT • Basis Sets Optimized for MC-DFT 28

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Heaven MC-DFT MC-ab initio MC-DFT 30

Acknowledgment Jien-Lian Chen Yi-Lun Sun Chun-Hui Li Ji-Ting Hong Kuo-Jui Wu Chen-Chang Wu NSC 31

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