Ivan Rostov, Australian National University, Canberra DFT Methods in Gaussian E-mail: Ivan.Rostov@anu.edu.au
Variety Of Methods In Computational Chemistry Cost CCSD(T) MP2 energy correction DFT methods Ab initio Hartree-Fock QM Semiempirical QM MM Force Fields Accuracy
Quality Size dependence • Ab initio MO Methods • CCSD(T) quantitative (1~2 kcal/mol) but expensive ~N6 • MP2 semi-quantitative and doable ~N4 • HF qualitative ~N2-3 • Density Functional Theory • DFT semi-quantitative and cheap ~N2-3 • Semi-empirical MO Methods • AM1, PM3, MNDO semi-qualitative ~N2-3 • Molecular Mechanics Force Field • MM3, Amber, Charmm semi-qualitative (no bond-breaking) ~N1-2 Variety of Methods in Computational Chemistry
Quantum Chemistry Basics Variational principle: Born-Oppenheimer (clamped-nuclei) approximation electrons are fast and moves in the field of fixed nuclei Hartree-Fock Approximation:
Density Functional Theory Basics Hohenberg-Kohn Theorems (1964) 1. Therefore, instead of Y dependent on 4N coordinates we would need just r0 dependent on just 3 coordinates 2. The variational principle for DFT If we would know how to express each of those four terms What about Tk[r] and Exc[r]? Thomas, Fermi (1927) poor accuracy, as was formulated for the uniform electron gas
Density Functional Theory Basics Kohn-Sham formalism resolves the problem with the kinetic energy term The big unknown left is The Hartree-Fock case:
Exchange functionals Ex[r] • Slater: ρ4/3 with theoretical coefficient a =2/3. Keyword: Used Alone: HFS, Comb. Form: S • Xαρ4/3 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.
Correlation functionals Ec[r] • 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.
Popular combinations of Ex[r] and Ec[r] • 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: • Exc = P2EXHF + P1(P4EXSlater + P3ΔExnon-local) + P6EClocal + P5ΔECnon-local • IOP(3/76),IOP(3/77) and IOP(3/78) setup P1 - P6 • B3LYP = • BLYP IOp(3/76=1000002000) IOp(3/77=0720008000) IOp(3/78=0810010000)
New functionals in revision E01 of G03 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
Time-Dependent DFT Runge-Gross theorem Runge-Gross equations: Linear response of the KS approximation
Time-Dependent DFT where
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
PyrroleRydberg States Basis set: aug-ccpvtz+R 14
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-separatedfunctionals 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
Asymptotic BehaviorofVxc(r). D.J. Tozer, N.C. Handy (1998) Must be: This is not observed for all model potentials listed earlier (exponential asymptotic) Solution (T. Yanai and K. Hirao group, 2004) First term goes to Ex, while second calculated together with J (T. Yanai, D.P.Tew, N.C.Handy, 2004) CAM-B3LYP: a = 0.19; b=0.46, m = 0.33
PyrroleRydberg States Basis set: aug-ccpvtz+R
Retinal Proteins Chromophores 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).
W. Koch, M.C. Holthausen, A Chemist’s Guide to Density Functional Theory (Wiley-VCH Verlag GmbH, 2001) M.E. Casida in Recent Advances in Density Functional Methods, Part 1 (World Scientific, Singapore, 1995) 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). Marques M.A.L. and Gross E.K.U. Annu. Rev. Phys. Chem55, 427 (2004). DFT references
Fujitsu Company for financial support and giving us opportunity to visit the beautiful country of Japan HiroHotta for constant help Professor ShinkohNanbu for invitation to the Kyushu University You, audience, for your attention Acknowledgements