Reactivity and Organocatalysis. (Adalgisa Sinicropi and Massimo Olivucci)

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1 Reactivity and Organocatalysis (Adalgisa Sinicropi and Massimo Olivucci)

2 The Aldol Reaction - O R 1 O R O O OH * * H R 2 R 1 R 2

3 The (1957) Zimmerman-Traxler Transition State Counterion (e.g. Li + ) Me R 1 R 2 H H R 1 δ- M M δ+ δ- δ+ O Me O or O O H H R 2 O OH * * R 1 R 2 Me Favoured Disfavoured

4 Aldolase I and II Enamine Me N R 1 M δ+ O δ- R 1 Enolate H H Me

5 The Intramolecular Proline Catalyzed Direct Asymmetric Aldol Hajos et al. (La Roche) & Eder et al. (Schering) CH 3 CN yield e.e. (95-100%) (90-96%)

6 The Intermolecular Proline Catalyzed Direct Asymmetric Aldol List et al.

7 The Catalyst Structure Requirements

8 The Catalyst Structural Requirements

9 The Mechanism O O N H O Zimmerman-Traxler type Nonacyclic TS

10 The Mechanism Water is not tollerated: unperturbed intramolecular hydrogen bonding through -COOH must be important.

11 The Stereoselectivity Two Zimmerman-Traxler-type Nonacyclic TS s: O O R Re face anti N OH H O H Si face R O sin N OH

12 Stereoselectivity

13 Stereoselectivity

14 The Curtin-Hammett Principle k A C K AB k B D C A B D ΔG TS A C A B TS B D C D Reaction Coordinate

15 The Curtin-Hammett Principle TS A C A B C TS B D confor. change D condensation

16 Enantioselectivity and Potential Energy Barriers The Curtin-Hammett principle states that: D:C = k B D [B]:k A C [A] = k B D [B]:(k A C /K AB )[B] = exp(δg B C -ΔG B D ) The rate constant expression is: k = kt h e ΔG RT

17 Enantioselectivity and Potential if ΔS B C ΔS B D then Energy Barriers ΔG B C -ΔG B D = ΔH B C -ΔH B D if ΔVol B C ΔVol B D then ΔH B C -ΔH B D = ΔU B C +PΔVol B C - ΔU B D -PΔVol B D = ΔU B C - ΔU B D

18 Enantioselectivity and Potential Energy Barriers ΔU = ΔV + Δ 3m 6 i 1 2 hν i if Δ B C 3m hν i Δ B D 1 2 hν i then 3m 6 i i ΔU B C - ΔU B D = V B C - V B D in conclusion: D:C = exp(v B C - V B D )

19 The Potential Energy Born-Oppenheimer Approximation Nuclear part of the Schrödinger Equation: [T n (R) + V nn (R) + E el (R)] χ = E tot χ Electronic part of the Schrödinger Equation: [T e (r) + V ee (r) + V en (r,r)] ψ = E el (R) ψ

20 The Potential Energy Surface of a Chemical Reaction [Tn(R) + Vnn(R) + Eel(R)] χ = Etot χ

21 Stationary Points

22 Stationary Points ω i = F ii µ i

23 Stationary Points

24 The Reaction Coordinate dr(s) ds = g(s) g(s)

25 Quantum Chemistry Molecular Mechanics V(R) = V nn (R) + E el (R) V m (R) = i j Z i Z j R ij Need Quantum Chemical (Electronic Structure) Methods

26 Quantum Chemical Technologies Semiempirical (ZINDO, AM1, MNDO, ) Computational Cost DFT (B3LYP, PBE, BPW91 ) Exact Ab initio (first principle) (CISD, CASSCF,, MRCI, FCI) CASPT2//CASSCF Error < 0.3 ev (excitation energies)

27 The ab-initio (Hartree-Fock based) Method E el Electronic Hamiltonian ( R) trial = ψ H ψ ψ ψ E el( R) true ψ = det The wavefunction The MO φ 1 ( 1)α( 1) φ 1 ( 1)β( 1)... φ n ( 1)β( 1) φ 1 ( 2)α( 2) φ 1 ( 2)β( 2)... φ n ( 2)β φ 1 ( n)α( n) φ 1 ( n)β( n)... φ n ( n)β n φ i = c ij ξ j j ( ) ( ) The basis set

28 The ab-initio (Hartree-Fock based) Method How to compute the orbitals efficiently? Fock operator f ˆ φ = ε φ k k k k f ˆ k = k Z i + 2J j k i r ki j ( ( ) K ( j k) ) Electron density k ρ = φ k 2

29 Density Functional Theory Hohenberg-Kohn existence theorem E el ( R) = 1 2 ψ 2 k ψ ρ( 1)ρ( 2) + i r ki i <j Z i ρ( 1) R ij + E xc The Generalized Gradient Approximation (GGA) E xc = ρ ri π ( ) 4 3 dτ i +Corr.Term( ρ)

30 (Hybrid) Density Functional Theory How to compute the electron density efficiently? The Kokn-Sham Operator (1965) F ˆ (r k ) = k ˆ F k η k = ε k η k Z i + J j (r k ) i r ki j +V xc V xc = Exc ρ The Kohn-Sham MO η k = c kj ξ j j The basis set

31 Geometry Optimization quadratic representation of the energy surface V(R) = V(R 0 )+ i V(R) R i R 0 (R i R i 0 )+ 1 2 i j 2 V(R) R i R j R o (R i R i 0 )(R j R j 0 ) Energy (value) Gradient (slope) t V(R) = V(R 0 )+ g R0 ΔR ΔRt H R0 ΔR Hessian (curvature)

32 Classification i ) Methods that use only the value of the function (i.e., in our case, the potential energy at R 0 ). One example is that of the Simplex method. ii) Methods that use the energy and the gradient of the function. For instance the steepest-descent and conjugated-gradient methods. iii) Methods that use the energy, first and second derivative of the function (e.g. in our case the potential energy its gradient and its hessian at R 0 ). An example is the Newton-Raphson method.

33 Steepest Descent Method Step Direction: V(R) b i = R 1 g Step Magnitude: ΔR = cg

34 Termination: multiple tresholds Maximum force V(R) R 1 max < t max RMS force i V(R) R i 2 < t RMS

35 The Newton-Raphson Method t V(R) = V(R 0 )+ g R0 ΔR ΔRt H R0 ΔR V(R) R = g R 0 ΔR = H 1 R0 g R0 t + ΔR t H R0 = 0 ΔR = dv dr d 2 V dr 2

36 The Newton-Raphson Method Min V(R) dv 1 ΔR = dr d 2 V dr R

37 The Newton-Raphson Method TS d 2 V dr 2 = 0 ΔR = dv dr d 2 V dr 2

38 Change in Curvature d 2V 2 =0 dr1 Max TS2 d 2V 2 =0 dr2 TS1 R1 TS3 R2 R1 R2

39 Trust Radius (Newton-Raphson) Method Hessian Normal modes ( ) R0 ΔR = H λi 1 g R0 Trust radius ΔR = τ 2 = i i Q i (Q i t g) R0 (Q i t g) 2 R 0 b i λ ( b i λ) 2 Force constants

40 Trust Radius (Newton-Raphson) Method λ < b 1 < b 2, b 3,, b 3m-6 b 1 < λ < b 2, b 3,, b 3m-6 look for a Min look for a TS b 1, b 2 < λ < b 3,, b 3m-6 look for a saddle point of index 2

41 Trust Radius (Newton-Raphson) Method

42 Quasi Newton-Raphson Methods The hessian is computationally very expensive QN Methods use Hessian updating formulas The most commonly used updating scheme in QN methods is based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) updating: H new = H old + ΔgΔgt H old ΔRΔR t H old ΔRΔg t ΔR t H old ΔR the BFGS updating produces positive definite hessians it cannot obviously be used when attempting a TS optimization. For this reason one has to use a different scheme such as the Murtaugh-Sargent (MS) formula

43 Optimization Efficiency