Quantum chemistry and wavefunction based methods for electron correlation!
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1 IMPRS Block Course Schmöckwitz, March 1, 2012 Quantum chemistry and wavefunction based methods for electron correlation Joachim Sauer Institut für Chemie, Humboldt-Universität HUMBLDT-UNIVERSITÄT ZU BERLIN Wlodzimierz Kolos
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3 C/Mg(001) Mg(001)/C Mg 2+ C 2- C/Mg(001) Example: C/Mg(001) bserved binding energy 15 kj/mol 2002 Nygren, Pettersson, J. Chem. Phys. 105 (1996) 9339
4 (terrace) Temperature programmed desorption (terrace) Wichtendahl,... Kuhlenbeck, Freund, Surf. Sci. 423 (1999) 90
5 Temperature programmed desorption 29 K peak: multilayer 76 K peak: defects Readhead 57 K Peak: {Mg 2+ } 5c (ν=10 13 s -1 ) 0.14 ev (15 kj/mol) Wichtendahl,... Kuhlenbeck, Freund, Surf. Sci. 423 (1999) 90 CH 4 /Mg(100) Monolayer, c 2x 2 R 45 o (Coulomb et al.) dipod configuration (Larese et al.) Tait, Dohnalek, Campbell, Kay, JCP 122 (2005) kj/mol (Θ=1) 13.1 kj/mol (40 K) (He scattering)
6 CH 4 /Mg(100) - Temperature Programmed Desorption (terrace) Attractive interaction between molecules Cluster formation at low coverage E 0 =11.1, γ =1.53 Tait, Dohnalek, Campbell, Kay, JCP 122 (2005) CH 4 /Mg(100) - Arrhenius Barrier vs. Desorption Energy E A = H d + RT H d (T)= E d + E ZPV + ΔH(T) E A = H d + RT kj/mol Arrhenius barrier E A T/K H d =E A - RT ΔH(T) H d (0) = H d - ΔH(T) ZPVE E d = H d (0) - ZPVE ± Vibrational contributions from PBE+D slab calculations
7 P. Dirac, Proc. Roy. Soc. (London) A123 (1929) 714: The general theory of quantum mechanics is now almost complete... The underlying physical laws necessary for the mathematical theory of... the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations that are much too complicated to be soluble. It therefore becomes desirable that approximate methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. The Royal Swedish Academy of Sciences has awarded The 1998 Nobel Prize in Chemistry in the area of quantum chemistry to Walter Kohn, University of California at Santa Barbara, USA and John A. Pople, Northwestern Univ., Evanston, Ill., USA (British citizen). Citation: "to Walter Kohn for his development of the density-functional theory and to John Pople for his development of computational methods in quantum chemistry."
8 Quantum chemistry Accuracy (system size) m constant(resources) Coupled cluster expansion of wave function: CCSD(T) 2nd order Moller-Plesset perturbation theory: MP2 Density functional theory (DFT) for full periodic structures feasible Dispersion largely missing: Van der Waals functional (Langreth & Lundquist) Pragmatic solution: DFT+D (e.g. Grimme) Surface Reaction TS Catalyst-Educt Complex Intrinsic Barrier Apparent Barrier Catalyst + Educt Binding energy Reaction energy Catalyst-Product Complex
9 DFT problems with barriers Van der Waals (dispersion) Catalyst-Educt Complex TS Intrinsic Barrier SI error (reaction site) Catalyst + Educt Binding energy Catalyst-Product Complex Zeolite catalysts: active sites (transition metal ions, protons) in a surface-only silica matrix Si Si Si Al M + -Si +Al, H Si H Al
10 Zeolite catalysis: Methanol-to-hydrocarbons Methanol from different sources MTG MT + CH CH 3 H 3 H CH 3 CH 3 Gasoline lefines C-C Formation, induction Methylation CH 2 =CH 2 CH 2 =CH-CH 3 Cyclic Polyens Hydrocarbon pool mechanism Aromatic HC Review: Stöcker, Microp. and Mesop. Mat. 29 (1999) 3-48 Haw et al, Kolboe et al. Methylation of ethene, propene, trans-2-butene in H-MFI Measured* Apparent Barrier Ethene 104 Propene 64 (-40) t-2-butene 40 (-24) Transition structure MeH (g) Alkene (g) MeH (ads) Alkene (g) Water (g) Alkene (g) MeH (ads) Alkene (ads) Water (ads) Alkene (ads) *S. Svelle, P.A. Ronning, S. Kolboe, J. Catal. 2004, 224, 115. S. Svelle, P.. Ronning, U. lsbye, S. Kolboe,J. Catal. 2005, 234, 385.
11 Methylation of alkenes in H-MFI - pbc DFT Error increases with chain length Svelle, Tuma, Rozanska, Kerber, Sauer, JACS 131 (2009) 816 Pragmatic solution: DFT + Dispersion (DFT+D) Many predecessors with DFT, HF+Disp, e.g. Ahlrichs and Scoles Many more sophisticated schemes E total = E DFT + E disp E disp = s 6 N 1 N C 6 ij 6 i =1 j =i +1 r ij f damp f damp (R) = 1 1+e -α (R/R 0-1) s 6 = 1.4/1.3/0.7 for BLYP/BP86/PBE Grimme, J. Comput. Chem., 2004, 25, 1463; 2006, 27, Parameters are as good for solids (condensed systems) as for molecules Note, however, Mg 2+ very different from Mg, whereas 2- similar to Implementation of Ewald sum for 1/r 6 for periodic systems Kerber, Sierka, Sauer, J. Comput. Chem. 29 (2008) 2088.
12 Methylation of alkenes in H-MFI - pbc DFT+D Dispersion - increases with chain length Svelle, Tuma, Rozanska, Kerber, Sauer, JACS 131 (2009) 816 Methylation of alkenes in H-MFI - pbc DFT+D Dispersion Too low barrier constant SIC error Svelle, Tuma, Rozanska, Kerber, Sauer, JACS 131 (2009) 816
13 Errors on energy barriers (Truhlar et al.) kcal/mol Nucleophilic substitution Reference data: CCSD(T) with extrapolation to complete basis set limit( W1 theory ) Divide and Conquer - Models and Methodsn _ a DFT (PBE/plane waves), VASP 20.2x20.5x13.5 Å; atoms Periodic Boundary Conditions CCSD(T)/cbs Atome C 3 11 Si 2 AlH 17 MLPR MP2/TZVP Atome C 3 72 Si 47 AlH 67 CC2 code b _
14 Hybrid high level : low level method Hybrid MP2(cluster):PBE+D(pbc) +ΔCCSD(T) method E hybrid (S,C) = E DFT+D (S) + [E MP2 (C) - E DFT+D (C)] High level correction Step 0: PBE+D optimization, periodic boundary conditions (pbc) Frequency calculation for stationary points, ZPVE Step 1: Hybrid MP2(cluster):PBE+D(pbc) optimization [Step 2: Basis set extrapolation to CBS limit, single point Step 3: CCSD(T)-MP2, small cluster model Tuma, Sauer, CPL 2004, 387, 388; PCCP, 2006, 8, 3955 Kerber, Sierka, Sauer, J. Comput. Chem. 2008, 29, 2088 Reuter/Scheffler, PRL 98 (2007) (C/Cu(111)) Stoll, JPC A 113 (2009) (Be, Mg crystals) Energy as functional of orbitals or electron density kinet. e-n e-e e-e en. attract Coulomb exchange/corr orbital density density E HF = E T + E N + E J + E Fock30 X (orbital) E DFT = E T + E N + E J + E XC (density) functional of density only E XC LDA = E X Dirac30 + E C VWN functional also of density gradient (Generalized Gradient Approximation) E XC BLYP = E X Dirac30 + E X B88 + E C LYP Self-interaction cancels
15 ρ(x) = N i=1 i(x) i*(x) Self-interaction correction Hartree-Fock E J + E X F30 (orbital) i = j r 12 ρ(x 2 )dx 1 dx i * (x 1 ) j (x 2 ) 1 2 ρ(x 1 ) 1 i, j 1 2 ij ij ij ji i, j 1 r 12 j (x 1 )i (x 2 )dx 1 dx 2 Coulomb and exchange terms cancel - self-interaction correction (SIC) Kohn-Sham E J + E XC (density) [ ] 1 2 ρ(x 1 ) 1 r 12 ρ(x 2 )dx 1 dx 2 + dxρ(x)v XC ρ, ρ 1 ii ii + i V 2 XC [ ρ, ρ] i 0 Coulomb and exchange do not cancel - self-interaction error * i * (x 1 ) j (x 2 ) ii ii ii ii = 0 1 r 12 i (x 1 ) j (x 2 )dx 1 dx 2 = ij ij Energy as functional of orbitals or electron density kinet. e-n e-e e-e en. attract Coulomb exchange/corr orbital density density E HF = E T + E N + E J + E Fock30 X (orbital) E DFT = E T + E N + E J + E XC (density) functional of density only E XC LDA = E X D30 + E C VWN functional also of density gradient (Generalized Gradient Approximation) E XC BLYP = E X Dirac30 + E X B88 + E C LYP orbital-density hybrid functional E XC B3LYP = a 0 E X F30 + (1- a 0 ) E X D30 + a X E X B88 +(1- a C ) E C VWN + a C E C LYP Self-interaction cancels Self-interaction cancels partially
16 Cluster anions (V 2 5 ) n- SM Size-dependent electron localization 20% 0% 50% Fock-exchange in functional CCSD(T) calculations confirm most stable structures C s - 2 A + 33 V D 2d - 2 B 1-57 kj/mol B3-LYP C s - 2 A CCSD(T)//BH-LYP BH-LYP C 2v - 2 A 2 V C s - 2 A - 48 kj/mol
![] -H + [Al 3 H] The hole is localized at one site (EPR -quartz) Solans-Monfort,](/docs-images/82/84911794/images/17-1.jpg "Branchadell, Sodupe, Sierka, Sauer, J Chem Phys 131 (2004) 6034 Embedded MFI models (5T,")
17 Electron hole defects in silica and zeolites Si 2 Al doped Si 2 electron hole self-trapped electron hole [Si 4 ] + [Al 4 ] -H + H-Zeolite [Al 3 H] + - e - - e - - e - [Si 4 ] [Al 4- ] -H + [Al 3 H] The hole is localized at one site (EPR -quartz) Solans-Monfort, Branchadell, Sodupe, Sierka, Sauer, J Chem Phys 131 (2004) 6034 Embedded MFI models (5T, 25T) - QM-Pot
18 Energy for electron hole creation (ev) BHLYP:Pot.fct. 5T 25T//5T Diff. Hybrid QM//Hybrid LR//Hybrid f LR//Hybrid (-1.27) (+0.45) corrected (7.99) aperiodic corr f=0.682 Systematic error of BHLYP (ev) G(d,p) cc-pvqz BHLYP, {T5} MFI 9.26 BHLYP, T1// {T5} MFI 9.47 CCSD(T), T1// {T5} MFI Increment
19 Expected rage of IP (ev) 5T// {25T} MFI BHLYP//Hybrid LR//Hybrid Hybrid aperiodic corr QM error [Al 3 H] [Si 2 ] Valence band edge (Si 2 ): ev Solans-Monfort, Branchadell, Sodupe, Sierka, Sauer, J Chem Phys 131 (2004) 6034
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