Quantum chemistry and wavefunction based methods for...

HUMBOLDT-UNIVERSITÄT ZU BERLIN

Quantum chemistry and wavefunction based methods for electron correlation !

IMPRS Block Course!Schmöckwitz, March 1, 2012!

Joachim Sauer!Institut für Chemie, Humboldt-Universität!

© Joachim Sauer, HU Berlin, 2012

Wlodzimierz Kolos 1928 - 1996




MgO(001)/CO!

Mg2+ C O

O2-

CO/MgO(001) !


Example: CO/MgO(001) Observed binding energy 15 kJ/mol!

CO/Mg(001) !

Nygren, Pettersson, J. Chem. Phys. 105 (1996) 9339!

2002


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Temperature programmed desorption!

Wichtendahl,... Kuhlenbeck, Freund, Surf. Sci. 423 (1999) 90!

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Temperature programmed desorption!

Wichtendahl,... Kuhlenbeck, Freund, Surf. Sci. 423 (1999) 90!

29 K peak: multilayer!76 K peak: defects!

Readhead!57 K Peak: {Mg2+}5c!

(ν=1013 s-1) ! !0.14 eV (15 kJ/mol)!


CH4/MgO(100)!

Tait, Dohnalek, Campbell, Kay, JCP 122 (2005) 164708!Monolayer,

„dipod� configuration (Larese et al.)

13.1 kJ/mol (40 K) (He scattering)

12.6 kJ/mol (Θ=1)!c 2x 2 R 45o (Coulomb et al.)


CH4/MgO(100) - Temperature Programmed Desorption!

Tait, Dohnalek, Campbell, Kay, JCP 122 (2005) 164708!

E0=11.1, γ =1.53

Attractive interaction between molecules!Cluster formation at low coverage

(terrace)!


CH4/MgO(100) - Arrhenius Barrier vs. Desorption Energy!

EA = Hd + RT! kJ!/mol!Arrhenius barrier EA! 12.6! 13.1!T/K! 47! 40!Hd=EA - RT! 12.2! 12.8!ΔH(T)1 1.1! 1.0!Hd(0) = Hd - ΔH(T)! 11.1! 11.8!ZPVE1 -4.2! -4.2!Ed = Hd(0) - ZPVE! 15.3! 16.0!

15.3± 0.6!

1 Vibrational contributions from PBE+D slab calculations!

EA = Hd + RT!Hd(T)= Ed + EZPV + ΔH(T)!


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."


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Accuracy · (system size)m ≈ constant(resources)!

Quantum chemistry

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)!

Coupled cluster expansion of wave function: CCSD(T)!2nd order Moller-Plesset perturbation theory: MP2 !


Surface Reaction!

TS

Catalyst!+ Educt!

Intrinsic!Barrier!

Catalyst-Educt!Complex!

Catalyst-Product!Complex!

Binding!energy!

Reaction energy!

Apparent!Barrier!


DFT problems with barriers!

TS

Catalyst!+ Educt!

Intrinsic!Barrier!

Catalyst-Educt!Complex!

Catalyst-Product!Complex!

Binding!energy!

Van der Waals (dispersion)

SI error (reaction site)


Zeolite catalysts: active sites (transition metal ions, protons) in a „surface-only� silica matrix!

H

-Si+Al,

AlSiO O

O O

AlSiO O

O O

H

SiSiO O

O OM+

H

-Si+Al,

AlSiO O

O O

AlSiO O

O O

H

SiSiO O

O OM+


Zeolite catalysis: Methanol-to-hydrocarbons!

Review: Stöcker, Microp. and Mesop. Mat. 29 (1999) 3-48!

Methanol from different sources !

CH3OH! CH3OCH3!

CH2=CH2!

CH2=CH-CH3!

Cyclic Polyens!

Aromatic HC!

Methylation!

C-C Formation, induction!

+ CH3OH!

Gasoline!Olefines!

MTG MTO!

Hydrocarbon pool!mechanism!

Haw et al, Kolboe et al.


MeOH (g) Alkene (g)

MeOH (ads) Alkene (g)

MeOH (ads) Alkene (ads) Water (ads)

Alkene (ads)

Water (g) Alkene (g)

Transition structure

*S. Svelle, P.A. Ronning, S. Kolboe, J. Catal. 2004, 224, 115.!S. Svelle, P.O. Ronning, U. Olsbye, S. Kolboe,J. Catal. 2005, 234, 385.!

Methylation of ethene, propene, trans-2-butene in H-MFI!

Measured* Apparent Barrier Ethene 104 Propene 64 (-40) t-2-Butene 40 (-24)


Methylation of alkenes in H-MFI - pbc DFT!

Svelle, Tuma, Rozanska, Kerber, Sauer, JACS 131 (2009) 816!

Error increases with chain length


s6 = 1.4/1.3/0.7 for BLYP/BP86/PBE

Parameters are as good for solids (condensed systems) as for molecules Note, however, Mg2+ very different from Mg, whereas O2- similar to O Implementation of Ewald sum for 1/r6 for periodic systems

Etotal = EDFT +Edisp

Edisp = −s6C6

ij

rij6j = i +1

N

∑i =1

N −1

∑ fdamp fdamp (R) = 1

1+e-α (R/R0 -1)

Pragmatic solution: DFT + Dispersion (DFT+D)!

Grimme, J. Comput. Chem., 2004, 25, 1463; 2006, 27, 1787.!

Many predecessors with DFT, HF+Disp, e.g. Ahlrichs and Scoles!Many more sophisticated schemes!

Kerber, Sierka, Sauer, J. Comput. Chem. 29 (2008) 2088.!


Methylation of alkenes in H-MFI - pbc DFT+D!


Dispersion - increases with chain length


Methylation of alkenes in H-MFI - pbc DFT+D!


Dispersion

Too low barrier!constant SIC error!


Errors on energy barriers (Truhlar et al.)!

Nucleophilic substitution

kcal/mol

Reference data: CCSD(T) with extrapolation to complete basis set limit(„W1 theory�)


DFT (PBE/plane waves), VASP 20.2x20.5x13.5 Å; 96 +18 atoms Periodic Boundary Conditions

b _

_a

CCSD(T)/cbs 17+17 Atome C3O11Si2AlH17 MOLPRO

MP2/TZVP 123+67Atome C3O72Si47AlH67 CC2 code

Divide and Conquer - Models and Methodsn!


Hybrid high level : low level method!

Ehybrid (S,C) = EDFT+D(S) + [EMP2(C) - EDFT+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) 176103 (CO/Cu(111)) !

Stoll, JPC A 113 (2009) 11483 (Be, Mg crystals)!

Hybrid MP2(cluster):PBE+D(pbc) +ΔCCSD(T) method!


Energy as functional of orbitals or electron density! kinet. e-n e-e e-e! en. attract Coulomb exchange/corr ! orbital density density !!!EHF = ET + EN + EJ + EX

Fock30 (orbital)!!

EDFT = ET + EN + EJ + EXC (density)!functional of density only!EXC

LDA = EXDirac30 + EC

VWN!

functional also of density gradient (Generalized Gradient Approximation)!EXC

BLYP = EXDirac30 + �EX

B88 + ECLYP!

Self-interaction cancels


12 ρ(x1) 1r12∫∫ ρ(x2 )dx1dx2 − 1

2 i*(x1) j∗(x2 )∫∫ 1

r12j (x1)i (x2 )dx1dx2

i, j∑ρ(x) =

i(x)i=1

N

∑ i*(x)

Coulomb and exchange terms cancel - self-interaction correction (SIC)!

Self-interaction correction!Hartree-Fock EJ + EX

F30 (orbital)

12 ij ij⎡⎣i, j∑ − ij ji ⎤⎦

i = j

Kohn-Sham ! EJ + EXC (density)!12 ρ(x1) 1r12∫∫ ρ(x2 )dx1dx2 + dxρ(x)VXC ρ,∇ρ[ ]∫

12 ii ii + i VXC ρ,∇ρ[ ] i ≠ 0

Coulomb and exchange do not cancel - self-interaction error!

* i*(x1) j∗(x2 )∫∫ 1

r12i (x1) j (x2 )dx1dx2 = ij ij

ii ii⎡⎣ − ii ii ⎤⎦ = 0


kinet. e-n e-e e-e! en. attract Coulomb exchange/corr ! orbital density density !!!EHF = ET + EN + EJ + EX

Fock30 (orbital)!!

EDFT = ET + EN + EJ + EXC (density)!functional of density only!EXC

LDA = EXD30 + EC

VWN!

functional also of density gradient (Generalized Gradient Approximation)!EXC

BLYP = EXDirac30 + �EX

B88 + ECLYP!

orbital-density hybrid functional!EXC

B3LYP = a0 EXF30 + (1- a0) EX

D30 + aX �EXB88 !

! +(1- aC ) ECVWN + aC EC

LYP!

Energy as functional of orbitals or electron density!

Self-interaction cancels

Self-interaction cancels partially


SOMO

20% 0% 50% !Fock-exchange in functional!

Cluster anions (V2O5)n-!

Size-dependent electron localization


V6O15- !

Cs- 2A�!-10!

C2v- 2A2!- 9!

- 48 kJ/mol !

BH-LYP!CCSD(T)//BH-LYP!B3-LYP!

- 57 kJ/mol !

Cs- 2A��!

D2d- 2B1!

Cs- 2A��!

V4O10- !

+ 33!

CCSD(T) calculations confirm most stable structures!


self-trapped electron hole

[AlO4•]

[AlO4-]

[SiO4•]+

[SiO4]

The hole is localized at one O site (EPR �-quartz)

electron hole

SiO2 Al doped SiO2

[AlO3•OH]+

[AlO3OH] -H+

-H+

- e-

H-Zeolite

- e- - e-

Electron hole defects in silica and zeolites

Solans-Monfort, Branchadell, Sodupe, Sierka, Sauer, J Chem Phys 131 (2004) 6034!


Embedded MFI models (5T, 25T) - QM-Pot!


Energy for electron hole creation (eV)!

BHLYP:Pot.fct.! 5T! 25T//5T! Diff.!

Hybrid! 7.40! 7.61!QM//Hybrid! 9.26! 8.81! -0.45!LR//Hybrid! -1.86! -1.20! +0.66!f·LR//Hybrid! (-1.27)! -0.82! (+0.45)!corrected! (7.99)! 7.99!+aperiodic corr.! 8.33!

f=0.682


Systematic error of BHLYP (eV)!

6-31++G(d,p)! cc-pVQZ!

BHLYP, {T5}MFI! 9.26!

BHLYP, T1// {T5}MFI! 9.47!

CCSD(T), T1// {T5}MFI! 9.60! 9.91-10.02!

Increment! 0.44-0.55!


Expected rage of IP (eV)!

5T// {25T}MFI!

BHLYP//Hybrid! 8.81! 8.81!LR//Hybrid! -1.20! -0.82!Hybrid! 7.61! 7.99!+ aperiodic corr.! 7.95! 8.33!+ QM error! 0.44! 0.55![AlO3

•OH]+! 8.4! 8.9![SiO2

•]+! 9.2! 9.5!

Valence band edge (SiO2): 10.2 - 10.6 eV!

Solans-Monfort, Branchadell, Sodupe, Sierka, Sauer, J Chem Phys 131 (2004) 6034!

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