macroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics

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1 Rate heory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van t Hoff & Arrhenius equation microscopic view (atomistic) statistical mechanics transition state theories Eyring theory effect of environment static: potential of mean force dynamic: Kramer s theory computing reaction rate optimizating transition states normal mode analysis simulating barrier crossing practical next week

2 Chromophore in water p-hydroxybenzylidene acetone (pck) CASSCF(6,6)/3-21G//SPCE molecular dynamics resonance structures

3 Rate of photoisomerization of double bond uni-molecular process, initiated by photon absorption

4 measuring reaction rate simulation & pump-probe fluorescence

5 kinetics & thermodynamics approaching equilibrium unimolecular process A k + k B d[a] dt d[b] dt = k + [A]+k [B] =+k + [A] k [B] conservation law so that d[a] dt [A]+[B] =[A] 0 = k + [A]+k ([A] 0 [A]) = (k + + k )[A]+k [A] 0 solution of the differential equations [A] = k + k +e k + + k (k +k )t [A] 0

6 kinetics & thermodynamics approaching equilibrium eventually... lim [A] = k t!1 k + + k [A] 0 lim [B] =[A] 0 [A] 1 = t!1 equilibrium constant & reaction free energy K = [B] 1 = k apple + G =exp [A] 1 k R k + k + + k [A] 0

7 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H S S = p = S = p G = H

8 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H S S = p = S = p G = p = H

9 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H S S = p = S = p G = p = p = H 2

10 temperature dependence of reaction rates Gibbs-Helmholtz relation G = H S S = p = S = p G = p = p = H 2

11 temperature dependence of reaction rates Van t Hoff equation equilibrium constant d ln K d ln K = = 1 R d d G R Gibbs-Helmholtz predicts effect of temperature on equilibrium constant G p = H R 2 d ln K d1/ = H R

12 temperature dependence of reaction rates Van t Hof equation equilibrium constant d ln K d ln K = = 1 R d d G R Gibbs-Helmholtz predicts effect of temperature on equilibrium constant K = k + k d G p d ln k + = H R 2 relation between equilibrium and rate constant d ln K d1/ = d d ln k = H R 2 H R

13 temperature dependence of reaction rates Van t Hof equation equilibrium constant d ln K d ln K = = 1 R d d G R Gibbs-Helmholtz predicts effect of temperature on equilibrium constant K = k + k d G p d ln k + = H R 2 relation between equilibrium and rate constant d ln K d1/ = d d ln k = H R 2 H R therefore d d1/ ln k + = E R ln k + =lna E R

14 temperature dependence of reaction rates Arrhenius equation activated state A A! B K = [A ] [A]

15 temperature dependence of reaction rates Arrhenius equation activated state A A! B K = [A ] [A] d ln K d1/ = H R k = a exp apple Ea R ln k =lna E a R 1

16 microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = R exp[ RB H]dpdq A exp[ H]dpdq = 1 k B

17 microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = Hamiltonian H = + V R exp[ RB H]dpdq A exp[ H]dpdq = 1 k B H = X i p 2 i 2m i + V (q 1,q 2,..,q n )

18 microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = Hamiltonian H = + V R exp[ RB H]dpdq A exp[ H]dpdq = 1 k B H = X i p 2 i 2m i + V (q 1,q 2,..,q n ) integrate over momenta R exp[ V ]dq K = RB A exp[ V ]dq equilibrium determined solely by potential energy surface

19 microscopic picture compute rates from simulations rare event rxn eq k =1/ rxn

20 microscopic picture compute rates from simulations rare event rxn eq k =1/ rxn basic assumptions initial rate stationary conditions d (p, q) =0 dt

21 microscopic picture compute rates from simulations rare event rxn eq k =1/ rxn basic assumptions initial rate stationary conditions flux d (p, q) dt =0 J = kc A c A = h (x x)i

22 microscopic picture compute rates from simulations rare event rxn eq k =1/ rxn basic assumptions initial rate stationary conditions flux d (p, q) dt =0 J = kc A c A = h (x sampling problem... (x )= x)i R exp[ V (x)] (x x )dx R exp[ V (x)]dx

23 Eyring theory assumptions classical dynamics no recrossing molecules at barrier in thermal equilibrium with molecules in reactant well

24 Eyring theory L observations barrier is flat: f(x )= du dx x=x =0

25 Eyring theory L observations barrier is flat: f(x )= du dx x=x =0 there are Nmolecules in L

26 Eyring theory L observations barrier is flat: f(x )= du dx x=x =0 there are Nmolecules in L reaction if v> L t

27 Eyring theory L observations barrier is flat: f(x )= du dx x=x =0 there are Nmolecules in L reaction if v> L t the number of molecules passing S in dt N rxn = N vdt L

28 Eyring theory L observations barrier is flat: f(x )= du dx x=x =0 there are Nmolecules in L reaction if v> L t the number of molecules passing S in dt N rxn = reaction rate N vdt L k + = N rxn Ndt = N N v L

29 Eyring theory L observations barrier is flat: f(x )= du dx x=x =0 there are Nmolecules in L reaction if v> L t the number of molecules passing S in dt N rxn = reaction rate N vdt L k + = N rxn Ndt = N N v L N N = q q A partition function k + = q q A vl

30 Eyring theory partition function of S q = 1 h L Z 1 1 exp[ ( p2 2m + V (x )]dp q = L h Z 1 1 exp[ p 2 2m ]dp exp[ V (x )] q = L h p 2mkB exp[ V (x ) k B ]

31 Eyring theory partition function of S q = 1 h L Z 1 1 exp[ ( p2 2m + V (x )]dp q = L h Z 1 1 exp[ p 2 2m ]dp exp[ V (x )] q = L h p 2mkB exp[ V (x ) k B ] only positive velocities contribute hv + i = hv + i = R 1 1 v (v)exp[ p 2 2m ]dp R 1 1 exp[ p 2 2m ]dp r 1 1 m 2 2mk B kb p 2mkB = 2 m

32 Eyring theory taking together to express rate k + = L h p 2mkB Lq A r kb 2 m exp apple V (x ) k B k + = k B hq A exp apple V (x ) k B

33 Eyring theory taking together to express rate k + = L h p 2mkB Lq A r kb 2 m exp apple V (x ) k B k + = k B hq A exp apple V (x ) k B partition function of A q A = 1 h Z x 1 exp apple V (x) k B dx Z 1 1 exp apple p 2 2m dp q A = 1 hp 2 mkb Z x 1 exp apple V (x) k B dx

34 Eyring theory taking together to express rate k + = L h p 2mkB Lq A r kb 2 m exp apple V (x ) k B k + = k B hq A exp apple V (x ) k B partition function of A q A = 1 h Z x 1 exp apple V (x) k B dx Z 1 1 exp apple p 2 2m dp q A = 1 hp 2 mkb Z x 1 exp apple V (x) k B dx

35 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 m!2 A(x x A ) 2! A = r kf m

36 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 m!2 A(x x A ) 2! A = r kf m partition function q A = hp 1 2 mkb s 2k B m! 2 A p q A = 1 h 2 k B 1! A

37 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 m!2 A(x x A ) 2! A = r kf m partition function q A = 1 h 2 k B 1! A Final result: Eyring equation k + =! A 2 exp apple V (x ) k B

38 Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 m!2 A(x x A ) 2! A = r kf m partition function q A = 1 h 2 k B 1! A Final result: Eyring equation attempt frequency k + =! A 2 exp apple V (x ) k B probability to be at barrier (Boltzmann factor)