Energy Profiles and Chemical Reactions

Transcription

1 Energy rofiles and Chemical eactions + B C D E egensburg er-la Norrby

2 Modeling Kinetics Molecular Modeling Stationary points Energies Barriers eaction rates DFT, ab initio, MM Ground & Transition states otential, Free eversible? ate constants, concentrations egensburg er-la Norrby

3 Energy The basic property The energy is a function of the geometry, that is, the coordinates (r). Each system strives to minimize the energy. Stable structures are located at energy minima. The force which acts on each atom to bring it to the minima is the slope of the energy function: F = - E/ r E r F The most common task is to move all atoms in the direction of the force to bring it to the minimum, where all other properties are calculated egensburg er-la Norrby

4 Molecular Energy Total energy: separate all electrons and nuclei completely tomization energy, heat of formation: separate into component atoms Bond energy: separate molecule into two fragments. otential energy: relative to an arbitrary state defined as zero Steric energy: relative to a hypothetical unstrained conformation. Conformational energy: relative to lowest energy conformation egensburg er-la Norrby

5 Calculating Energies Target accuracy: 2-4 kj/mol Quantum Mechanics Solving the Schrödinger equation The wavefunction: the position of every electron ny property available from the wavefunction Tradeoff between accuracy and speed Currently ca 200 atoms with acceptable accuracy Total energies: E = 0 at total separation Molecular Mechanics Simple, analytic functions, atomic positions only Known systems only Very fast, millions of atoms possible Steric energies: E = 0 for unstrained structures. nly conformational energies valid ˆ H " = E " E = # k( l " l 0 ) 2 l egensburg er-la Norrby

6 Structures Energy minima, conformations Energy!E!E Transition state, reaction rates Minimum, observable egensburg er-la Norrby

7 opulation distributions Boltzmann factor p 1 =12%!E 1 =4 kj/mol T=300K (27 C) p 0 =61% p 2 =27%!E 2 =2 kj/mol p 1 p 0 = e!e 1/T (cf. G = TlnK) p 0 = ules of thumb for ratios: (also valid for relative rates) e!e 0/T e!e 0/T + e!e 1/T + e!e 2/T t 25 C: 2/1, 2 kj/mol 5/1, 4 kj/mol 10/1, 6 kj/mol (1.4 kcal/mol) 4 kj/mol t -78 C, 12/1 t 25 C, 5/1 t 75 C, 4/1 t 250 C, 5/2 egensburg er-la Norrby

8 Calculated energies The basic calculated energy is the potential energy, varying as a function of the geometry E Vibrational levels ZE U T (!H T ) E 0 t 0 K, all molecules are at the lowest vibrational level, G 0 = H 0 = U 0 = E 0 +ZE t higher temperatures, higher levels are populated; H T > H 0 Experiments depend on the free energy, which also includes entropy, G = H - TS egensburg er-la Norrby

9 Degenerate conformations Conformational entropy In the example below, there is one global minimum and two equivalent (experimentally indistinguishable) higher conformations. gauche anti gauche 2 kj/mol 2 kj/mol From the rules of thumb, at room temperature the ratios are 1:2:1. That means that there are equal amounts of gauche and anti, that is, the total free energy of gauche and anti are equal. The free energy of gauche is lowered by a conformational entropy contribution equal to ln2. The entropy contribution is always lnw, where W is the number of states. egensburg er-la Norrby

10 Quizz The global minimum,, has a single conformation. The next lowest conformation, B, 2 kj/mol higher than, has 10 identical (degenerate) copies. How much lower is the free energy of B compared to? a) 4 kj/mol b) 6 kj/mol c) 8 kj/mol d) 10 kj/mol egensburg er-la Norrby

11 Entropy and free energy G = H T S The entropy, S = lnw, just comes from the fact that there are many possible states in each conformation. G could be obtained equivalently by summing the energy over all possible states, but it is easier to calculate an energy minimum once, and then get the entropy for that minimum by counting the number of equivalent conformations, the population in all vibrational states, etc. Higher energy, but more possible states, therefore higher population and thus lower free energy egensburg er-la Norrby

12 Gas phase free energy G tot = E 0 + ZE + H vib + T S vib + G rot/trans + T S conf Temperature independent: E 0 ZE The potential energy at the calculated minimum Zero point energy Temperature dependent (no contribution at 0 K): H vib From averaging over vibrational levels (Note: many packages include ZE in this term!) S vib Vibrational entropy G rot/trans Contribution from translation and rotation ule of thumb: 30±10 kj/mol for each molecule at room temperature S conf Conformational entropy. Either ln(number of conf.) or sum over all conformations, including degenerate ones. egensburg er-la Norrby

13 Solvent Charge stabilization Charges, especially localized anions, are much more stable in solvent. F Extremely strong base in gas phase, less so in solvent Delocalized charge, less difference between gas phase and solvent. bout equal to fluoride in water. Charge separation is almost impossible in gas phase gas phase H 3 N water H 2 N H egensburg er-la Norrby

14 Solvation Gas phase H 3 N Born: Free energy gain from exposed surface of polar atoms, SM2, SM3,... Continuum: cavity with compensating surface charges CM, CSM!"!"!"!" H 3 N!"!"!+!+!+!+!+!+!+!+ N H 2 H H 3 N Explicit solvation (QM/MM) Microsolvation H 3 N H 3 N egensburg er-la Norrby

15 Transition state energies and rates Calculated from free energies in solvent, G! roduct!g eactant!g B!G!G B!!G roduct B roduct ratio at equilibrium: "##G T K = e ate constant, formation of : k = " k b T h e#$g T Kinetic ratio, /B: k k B = e "##G T egensburg er-la Norrby

16 Two reaction steps eactant Intermediate roduct B B = tot First step: fast, exergonic Low T: I II can be studied independently I verall barrier last step, G = G B G II!G II!G tot!g B Common state diagram: G B min(g I,G II ) I II!G B B tot First step: endergonic Intermediate II cannot be seen Curtin-Hammett!!G!G tot!g B verall barrier G = G B G I egensburg er-la Norrby

17 Curtin-Hammett derivation I k a k a II B k b B Steady state: d[ II] = k a [ I]" ( k "a + k b )[ II] = 0 dt " [ II] = k a[ I ] k #a + k b rate = k b [ II] = k a k b [I] # k obs = k a k b k "a + k b k "a + k b k a = ce " ( G "G I ) T k "a = ce " ( G "G II ) T ( k obs = c e" G "G I ) T e " ( G B "G II ) T e " ( G "G II ) T + e " ( G B "G II ) = c e" GB "GI T 1+ e " G B "G k b = ce " ( G B "G II ) T ( ) T ( ) T # ( c = #k b T h) % G B > G : $G obs = G B " G I ' & G B = G : half rate ' ( G B < G : $G obs = G " G I Intermediate II is kinetically insignificant (as long as it is higher in energy than I)! TS is kinetically insignificant as long as it is significantly lower than TS B egensburg er-la Norrby

18 re-equilibria, TS conformations Free energy profile State diagram G 2 G 3 G TS "G = "T ln i T # $ min G i G TS i e ( ) G 1 G 2 G 3 G react "G G react = "T ln i T # $ min G i i e ( ) G prod Ensemble free energies always slightly lower than the lowest single free energy arallel paths can be seen as a TS ensemble (TS conformations) Note that serial TSs have an apparent barrier slightly higher than the highest single TS (cf. Curtin-Hammett) egensburg er-la Norrby

19 Multi-step reactions Effectively irreversible step (EIS): ny barrier higher than all subsequent barriers, C, and D, but not B (below) nly EIS can be kinetically significant basin I basin II B C basin III D D IV Barriers are calculated from the preceding ensemble (basin) Virtually the same as calculate from the lowest preceding point without passing an EIS ny EIS can be product-controlling (i.e., selectivity-determining). The rate determining step (rds) is the step with the highest barrier The resting state is the basin preceding the rds. For an alternative nomenclature, see: Kozuch, Shaik, cc. Chem. es. 2011, 44, 101 C egensburg er-la Norrby

20 Multi-step reactions Effectively irreversible step (EIS): ny barrier higher than all subsequent barriers, C, and D, but not B (below) nly EIS can be kinetically significant State diagram: I C D II III D IV Barriers are calculated from the preceding ensemble (basin) Virtually the same as calculate from the lowest preceding point without passing an EIS ny EIS can be product-controlling (i.e., selectivity-determining) The rate determining step (rds) is the step with the highest barrier (II C) The resting state is the basin preceding the rds (II) For an alternative nomenclature, see: Kozuch, Shaik, cc. Chem. es. 2011, 44, 101 C egensburg er-la Norrby

21 eaction progress Initial phase: starting material I is converted to intermediate II t low T, step can be studied independently, and the product is II Steady state: intermediate II is reacting slowly to product IV. C I D II III D IV t the steady state, the rate across every EIS is equal: only II and IV detectable spectroscopically If any step requires an external reagent, the rate is dependent also on that concentration slow addition can change the qualitative character of the state diagram Example: r X dl 2 r X d L L r' Sn 3 r r' d L L C k [ I] = k C [ II] = k D [ III] r r' egensburg er-la Norrby

22 Catalytic cycles In catalytic cycles at steady state, the starting point is arbitrary The nature of any potential EIS must be checked across the boundary C I II III D D I IV C I II D is not an EIS, it is lower than the subsequent D is reversible, and kinetically insignificant D III C egensburg er-la Norrby

23 Catalytic cycles In catalytic cycles at steady state, the starting point is arbitrary The nature of any potential EIS must be checked across the boundary C III II C D I III II C D is not an EIS, it is lower than the subsequent D is reversible, and kinetically insignificant The state diagram does not include I or D Even the nature of C as an EIS is doubtful If C is lower than, then the rds is II D can potentially be made significant again by slow addition egensburg er-la Norrby

24 Modeling and kinetics Current DFT methods can calculate relative barriers to within kj/mol Several potential pitfalls, not always sufficient for determination of rds I ' C C C' III D D D' I' II III' IV II' IV' Calculation of the same type of barrier after a small perturbation Generally accurate to 1-2 kj/mol Typical perturbations: Substituents (Hammett studies) Isotopic substitution (kinetic isotope effects) Evaluate in state diagram, relevant TS - lowest preceding point egensburg er-la Norrby

25 Cu-catalyzed cyclopropanation CEt rate limiting stereoselectivity Hammett isotope effect N 2 N 2 CEt tbu tbu Cu N N Cu N N tbu tbu r r CEt tbu Cu N N tbu tbu Cu N N product determining CEt tbu tbu Cu N N tbu egensburg er-la Norrby

26 The Heck reaction catalytic cycle with several potential EIS L d L r X d L L r d L L r d L L H r d L L r H d L L L d L ryl halide r X Hammett xidative addition ath Ligand exchange Migratory insertion egio- and stereoselectivity Bond rotation Double-bond regioselectivity r Hammett h Me h!-hydride elimination Me Dissociation Deprotonation eversibility h h h egensburg er-la Norrby

27 Quizz: Ni-catalyzed Heck reaction Ni h Tf h Ni h Ni h H Ni Cy 2 NEt h Ni What step is product-determining? Vote for i-iv i: oxidative addition ii: migratory insertion iv: deprotonation iii:!-hydride elimination Ns Tf + Et Ns Et + Tf Et egensburg er-la Norrby

28 Quizz: Ni-catalyzed Heck reaction Ni h Tf h Ni h Ni h H Ni Cy 2 NEt h Ni What step is product-determining? Vote for i-iv i: oxidative addition ii: migratory insertion iv: deprotonation iii:!-hydride elimination Tf + + egensburg er-la Norrby

29 Quizz: Ni-catalyzed Heck reaction Ni h Tf h Ni h Ni h H Ni Cy 2 NEt h Ni What step is product-determining? Vote for i-iv i: oxidative addition ii: migratory insertion iv: deprotonation iii:!-hydride elimination Tf + CN CN + CN egensburg er-la Norrby

30 The asymmetric HWE reaction H + H X H + H X racemic mixture Step 1 (') 2 CH 3 C(CH 3 ) 2 h High reagent selectivity ensures (2S) intermediate. Low diastereoselectivity for C(3) and C(4). H (') 2 (S) () C* H (') 2 (S) () C* H (') 2 (S) (S) C* H (') 2 C* () (S) () (S) H H H H X X X X FE anti-fe anti-fe FE (S) (S) Step 2 High diastereoselectivity for C(3) and C(4) C* C* *C *C X minor X major enaniodivergent products X X minor egensburg er-la Norrby

31 HWE reaction selectivity filtering (') 2 C* (') 2 C* X (SSS) X Filter (S) (SSS) (SS) Filter (SS) (S) (SS) (SS) Z (mostly ) E (mostly S) egensburg er-la Norrby

32 Summary Modeling yields reliable structures, sometimes K energies The entire reaction path should be evaluated Find the effectively irreversible steps nly these can control selectivity Barrier evaluated from lowest point in the preceding basin (Curtin-Hammett) Comparison of different types of TS sometimes unreliable Comparing perturbations for similar steps gives high accuracy egensburg er-la Norrby