Ivan Rostov, Australian National University, Canberra

Cost Accuracy MM Force Fields Semiempirical QM Ab initio Hartree-Fock QM MP2 energy correction CCSD(T) DFT methods 2

QualitySize dependence  Ab initio MO Methods  CCSD(T)quantitative (1~2 kcal/mol) but expensive~N 6  MP2semi-quantitative and doable~N 4  HFqualitative~N 2-3  Density Functional Theory  DFTsemi-quantitative and cheap~N 2-3  Semi-empirical MO Methods  AM1, PM3, MNDO semi-qualitative~N 2-3  Molecular Mechanics Force Field  MM3, Amber, Charmmsemi-qualitative (no bond-breaking) ~N 1-2 3

Variational principle: Hartree-Fock Approximation: Born-Oppenheimer (clamped-nuclei) approximation electrons are fast and moves in the field of fixed nuclei 4

Hohenberg-Kohn Theorems (1964) 2. The variational principle for DFT If we would know how to express each of those four terms Thomas, Fermi (1927) 1. What about T k [  ] and E xc [  ]? 5 Therefore, instead of  dependent on 4N coordinates we would need just  0 dependent on just 3 coordinates poor accuracy, as was formulated for the uniform electron gas

Kohn-Sham formalism resolves the problem with the kinetic energy term The Hartree-Fock case: The big unknown left is 6

Slater: ρ 4/3 with theoretical coefficient  =2/3. Keyword: Used Alone: HFS, Comb. Form: S Xαρ 4/3 with the empirical coefficient of 0.7, usually used when this exchange functional is used without a correlation functional Keyword: Used Alone: XAlpha, Comb. Form: XA. Becke 88: Becke's 1988 functional, which includes the Slater exchange along with corrections involving the gradient of the density. Keyword: Used Alone: HFB, Comb.Form: B. Perdew-Wang 91: The exchange component of Perdew and Wang's 1991 functional. Keyword: Used Alone: N/A, Comb. Form: PW91. Modified PW91: as modified by Adamo and Barone. Keyword: Used Alone: N/A, Comb. Form: MPW. Gill 96: The 1996 exchange functional of Gill. Keyword: Used Alone: N/A, Comb. Form: G96. PBE: The 1996 functional of Perdew, Burke and Ernzerhof. Keyword: Used Alone: N/A, Comb. Form: PBE. OPTX: Handy's OPTX modification of Becke's exchange functional. Keyword: Used Alone: N/A, Comb. Form: O. TPSS: The exchange functional of Tao, Perdew, Staroverov, and Scuseria. Keyword: Used Alone: N/A, Comb. Form: TPSS. 7

VWN: Vosko, Wilk, and Nusair 1980 correlation functional fitting the RPA solution to the uniform electron gas, often referred to as Local Spin Density (LSD) correlation. VWN V(VWN5): Functional which fits the Ceperly-Alder solution to the uniform electron gas. LYP: The correlation functional of Lee, Yang, and Parr which includes both local and non-local terms. PL (Perdew Local): The local (non-gradient corrected) functional of Perdew (1981). P86 (Perdew 86): The gradient corrections of Perdew, along with his 1981 local correlation functional. PW91 (Perdew/Wang 91): Perdew and Wang's 1991 gradient-corrected correlation functional. B95 (Becke 95): Becke's τ-dependent gradient-corrected correlation functional (defined as part of his one parameter hybrid functional. PBE: The 1996 gradient-corrected correlation functional of Perdew, Burke and Ernzerhof. TPSS: The τ-dependent gradient-corrected functional of Tao, Perdew, Staroverov, and Scuseria. 8

SVWN=LSDA SVWN5 BLYP Hybrid functionals B3LYP B3P86, B3PW91, B1B95 (1 parameter), B1LYP, MPW1PW91, B98, B971, B972, PBE1PBE etc. You can even construct your own. Gaussian provides such a functionality: E xc = P 2 E X HF + P 1 (P 4 E X Slater + P 3 ΔE x non-local ) + P 6 E C local + P 5 ΔE C non-local IOP(3/76),IOP(3/77) and IOP(3/78) setup P 1 - P 6 B3LYP = BLYP IOp(3/76= ) IOp(3/77= ) IOp(3/78= ) 9

Hybrid M05, MO5-2X with the same parameterization scheme but different set of parameters (25!) As reported by Donald Truhlar and Yan Zhao, M05 and MO5-2X outperform other parameterized hybrid functionals in nonmetallic thermochemical kinetics, thermochemistry and noncovalent interactions. MO5-2X is especially good for calculation of the bond dissociation energies, stacking and hydrogen-bonding interactions in nucleobase pairs 10

Runge-Gross theorem Runge-Gross equations: Linear response of the KS approximation 11

where 12

 Time Dependent DFT (TD-DFT) is widely used to calculate molecular electronic excitation energies.  Sufficiently accurate to be useful  Sufficiently economical to apply to large molecules  Not as accurate as highly correlated methods such as CASPT2 or CC3  Problems with Rydberg and Charge Transfer States, double excitations, intensities

14 CASPT2B3LYPPBE0 A B A B B B A A B A MAE Basis set: aug-ccpvtz+R 14

15 CASPT2B3LYP W1 A’’ W2 A’’ NV1 A’ NV2 A’ CT1 A’ CT2 A’’

16 Problems with Rydberg and Charge Transfer States are due to incorrect long range potentials (also problems with the response kernel, self-interaction) Standard DFT functionals are too short range Modifications to the long range part of the exchange potential are needed One approach is to use range-separated functionals constructed from different short range (high density) and long range (low density) forms. Short and long range components evaluated using different techniques Other approaches are possible e.g. orbital dependent potentials

Must be: D.J. Tozer, N.C. Handy (1998) This is not observed for all model potentials listed earlier (exponential asymptotic) Solution (T. Yanai and K. Hirao group, 2004) CAM-B3LYP:  = 0.19;  =0.46,  = 0.33 (T. Yanai, D.P.Tew, N.C.Handy, 2004) First term goes to E x, while second calculated together with J 17

CASPT2B3LYPPBE0CAM- B3LYP A B A B B B A A B A MAE Basis set: aug-ccpvtz+R 18

19 CASPT2B3LYPHCTH-ACCAM- B3LYP W1 A’’ W2 A’’ NV1 A’ NV2 A’ CT1 A’ CT2 A’’

MethodBasis S1S1 S2S2 EfEf Experiment (in methanol)2.79, 2.82 Experiment – solvent blue shift2.41,2.43 CASPT2(12,12) G(d) TD-BP866-31G+(d) TD-B3LYP6-31G(d) TD-B3LYP6-31G+(d) TD-B3LYPSV(P) TD-CAMB3LYP6-31G(d) TD-CAMB3LYP6-31G+(d) TD-CAMB3LYPSV(P) TD-CAMB3LYPTZV(P) Table 1. Excitation energies (eV) and oscillator strengths for 6-cis-11-cis PSB11. 1 CASPT2 calculations performed on geometry optimized with the state averaged CAS(12,12)/6-31G(d). All TD-DFT calculations employed geometry optimized with B3LYP/6-31G(d). 20

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1. W. Koch, M.C. Holthausen, A Chemist’s Guide to Density Functional Theory (Wiley- VCH Verlag GmbH, 2001) 2. M.E. Casida in Recent Advances in Density Functional Methods, Part 1 (World Scientific, Singapore, 1995) 3. M.E. Casida in Recent Developments and Applications of Modern Density Functional Theory, Theoretical and Computational Chemistry, vol 4., ed. by J.M. Seminario (Elsevier, Amsterdam, 1996). 4. Marques M.A.L. and Gross E.K.U. Annu. Rev. Phys. Chem 55, 427 (2004). 24

 Fujitsu Company for financial support and giving us opportunity to visit the beautiful country of Japan  Hiro Hotta for constant help  Professor Shinkoh Nanbu for invitation to the Kyushu University  You, audience, for your attention 25