Rate Theory (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
Chromophore in water p-hydroxybenzylidene acetone (pck) CASSCF(6,6)/3-21G//SPCE molecular dynamics resonance structures
Rate of photoisomerization of double bond uni-molecular process, initiated by photon absorption
Rate of photoisomerization of double bond uni-molecular process, initiated by photon absorption
measuring reaction rate simulation & pump-probe fluorescence
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 [A]+[B] =[A] 0 d[a] dt = k + [A]+k ([A] 0 [A]) = (k + + k )[A]+k [A] 0 solution of the differential equations [A] = k + k + e (k +k )t k + + k [A] 0
kinetics & thermodynamics approaching equilibrium eventually... lim [A] = t k k + + k [A] 0 lim [B] =[A] 0 [A] = t equilibrium constant & reaction free energy K = [B] = k + =exp G [A] k RT k + k + + k [A] 0
temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T
temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T G T T T p = H T
temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T G G T T T p = H T T T p = H T 2
temperature dependence of reaction rates Gibbs-Helmholtz relation G = H TS S = H G T G T G p = S = G H T T p G T = H T G G T T T p = H T T T p = H T 2
temperature dependence of reaction rates Van t Hoff equation equilibrium constant d ln K dt ln K = G = 1 R d dt RT Gibbs-Helmholtz predicts effect of temperature on equilibrium constant G T p = H RT 2 d ln K d1/t = H R
temperature dependence of reaction rates Van t Hof equation equilibrium constant d ln K dt ln K = G = 1 R d dt RT Gibbs-Helmholtz predicts effect of temperature on equilibrium constant K = k + k G T p = H RT 2 relation between equilibrium and rate constant d ln K d dt ln k + d dt ln k = H RT 2 d1/t = H R
temperature dependence of reaction rates Van t Hof equation equilibrium constant d ln K dt ln K = G = 1 R d dt RT Gibbs-Helmholtz predicts effect of temperature on equilibrium constant K = k + k G T p = H RT 2 relation between equilibrium and rate constant therefore d d1/t ln k = E R + a d ln K d dt ln k + d dt ln k = H RT 2 d1/t = H R
temperature dependence of reaction rates Arrhenius equation activated state A A B K = [A ] [A]
temperature dependence of reaction rates Arrhenius equation activated state A A B K = [A ] [A] d ln K d1/t = H R k = a exp E a RT ln k =lna E a R 1 T
microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = B exp[ βh]dpdq A exp[ βh]dpdq β = 1 k B T
microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = Hamiltonian H = T + V B exp[ βh]dpdq A exp[ βh]dpdq β = 1 k B T H = i p 2 i 2m i + V (q 1,q 2,..,q n )
microscopic picture statistical mechanics partition function K = p B p A = Q B Q A = Hamiltonian H = T + V B exp[ βh]dpdq A exp[ βh]dpdq β = 1 k B T H = i p 2 i 2m i + V (q 1,q 2,..,q n ) integrate over momenta K = exp[ βv ]dq B exp[ βv ]dq A equilibrium determined solely by potential energy surface
microscopic picture compute rates from simulations rare event τ rxn τ eq k =1/τ rxn
microscopic picture compute rates from simulations rare event τ rxn τ eq basic assumptions initial rate stationary conditions dρ(p, q) =0 dt k =1/τ rxn
microscopic picture compute rates from simulations rare event τ rxn τ eq basic assumptions initial rate stationary conditions flux dρ(p, q) dt =0 J = kc A c A = Θ(x x) k =1/τ rxn
microscopic picture compute rates from simulations rare event τ rxn τ eq basic assumptions initial rate stationary conditions flux dρ(p, q) dt =0 k =1/τ rxn J = kc A c A = Θ(x x) sampling problem... ρ(x exp[ βv (x)]δ(x x )dx )= exp[ βv (x)]dx
Eyring theory assumptions classical dynamics no recrossing molecules at barrier in thermal equilibrium with molecules in reactant well
Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0
Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl
Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are reaction if δnmolecules in v> δl δt δl
Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl reaction if v> δl δt the number of molecules passing TST in dt N rxn = δn vdt δl
Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl reaction if v> δl δt the number of molecules passing TST in dt N rxn = δn vdt δl reaction rate k + = N rxn Ndt = δn N v δl
Eyring theory δl observations barrier is flat: f(x )= du dx x=x =0 there are δnmolecules in δl reaction if v> δl δt the number of molecules passing TST in dt N rxn = δn vdt δl reaction rate k + = N rxn Ndt = δn N v δl δn N = q q A partition function k + = q q A v δl
Eyring theory partition function of TST q = 1 h δl exp[ β( p2 2m + V (x )]dp q = δl h exp[ β p2 2m ]dp exp[ βv (x )] q = δl h 2mkB T π exp[ V (x ) k B T ]
Eyring theory partition function of TST q = 1 h δl exp[ β( p2 2m + V (x )]dp q = δl h exp[ β p2 2m ]dp exp[ βv (x )] q = δl h 2mkB T π exp[ V (x ) k B T ] only positive velocities contribute v + = v + = vθ(v)exp[ β p2 2m ]dp exp[ β p2 2m ]dp 1 1 m 2 2mk BT kb 2mkB T π = T 2πm
Eyring theory taking together to express rate k + = δl h 2mkB T π δlq A kb T 2πm exp V (x ) k B T k + = k BT hq A exp V (x ) k B T
Eyring theory taking together to express rate k + = δl h 2mkB T π δlq A kb T 2πm exp V (x ) k B T k + = k BT hq A exp V (x ) k B T partition function of A q A = 1 h x exp V (x) k B T dx exp β p2 2m dp q A = 1 h 2πmkB T x exp V (x) k B T dx
Eyring theory taking together to express rate k + = δl h 2mkB T π δlq A kb T 2πm exp V (x ) k B T k + = k BT hq A exp V (x ) k B T partition function of A q A = 1 h x exp V (x) k B T dx exp β p2 2m dp q A = 1 h 2πmkB T x exp V (x) k B T dx
Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m
Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m partition function q A = h 1 2πmkB T 2k B T mω 2 A π q A = 1 h 2πk BT 1 ω A
Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m partition function q A = 1 h 2πk BT 1 ω A Final result: Eyring equation k + = ω A 2π exp V (x ) k B T
Eyring theory harmonic approximation V (x) 1 2 k f (x x A ) 2 V (x) 1 2 mω2 A(x x A ) 2 ω A = kf m partition function q A = 1 h 2πk BT 1 ω A Final result: Eyring equation attempt frequency k + = ω A 2π exp V (x ) k B T probability to be at barrier (Boltzmann factor)
static solvent effects: potential of mean force oops
Dynamic solvent effects: Kramers Theory coupling between reaction coordinate and other coordinates friction due to interactions dv dt = g v = γv m v = v 0 e γt thermal noise (Brownian motion): Langevin dynamics dv dt = γv + F R(t) noise properties F R =0 F R (t 1 )F R (t 2 ) = φ(t 2 t 1 ) fδ(t 2 t 1 ) solution v = v 0 exp [ γt]+exp[ γt] t 0 exp [γt] F R (τ)dτ
Dynamic solvent effects: Kramers Theory coupling between reaction coordinate and other coordinates solution v = v 0 exp [ γt]+exp[ γt] properties v = v 0 exp [ γt] t 0 exp [γt] F R (τ)dτ v 2 = v 2 0 exp [ 2γt]+ f 2γ (1 exp [ 2γt]) lim v =0 t lim t v2 = f 2γ equipartition theorem: 1 2 mv2 = 1 2 k BT F R (t)f R (0) = δ(t)2γk B T/m f =2γk B T/m
Dynamic solvent effects: Kramers Theory Fokker-Planck equation P (r, v; t) t = v P (r, v; t) r + 1 M U r P (r, v; t) v + γ v (vp(r, v; t)) + γk BT M 2 P (r, v; t) v 2 probability P (r, v; t)drdv to find a particle at r, r + dr with velocity v, v + dv
Dynamic solvent effects: Kramers Theory Fokker-Planck equation P (r, v; t) t = v P (r, v; t) r + 1 M U r P (r, v; t) v + γ v (vp(r, v; t)) + γk BT M 2 P (r, v; t) v 2 stationary solution P t =0 mv 2 P (r, v) = 1 Q exp 2 + V (r) /k B T Boltzmann distribution
Dynamic solvent effects: Kramers Theory Fokker-Planck equation P (r, v; t) t = v P (r, v; t) r + 1 M U r P (r, v; t) v + γ v (vp(r, v; t)) + γk BT M 2 P (r, v; t) v 2 steady state solution mv 2 P (r, v) =Y (r, v) 1 Q exp boundary conditions 2 + V (r) /k B T r r A Y (r, v) =1 r r C Y (r, v) =0
Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2
Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to Fokker-Planck equation k + = ω A γ 2 2πω B 4 + ω2 B γ exp [ (U(r B ) U(r A )) /k B T ] 2
Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to F-P equation γ 2 k + = ω A 2πω B 4 + ω2 B γ 2 exp [ (U(r B ) U(r A )) /k B T ]
Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to F-P equation γ 2 k + = ω A 2πω B 4 + ω2 B γ 2 exp [ (U(r B ) U(r A )) /k B T ] limiting cases γ/2 ω B k + = ω Aω B 2πγ exp U /k B T γ/2 ω B k + = ω A 2π exp U /k B T high friction low friction
Dynamic solvent effects: Kramers Theory free energy surface surface U(r) =U(r A )+ 1 2 mω2 A(r r A ) 2 U(r) =U(r B ) 1 2 mω2 B(r r B ) 2 stationary solution to F-P equation γ 2 k + = ω A 2πω B 4 + ω2 B γ 2 exp [ (U(r B ) U(r A )) /k B T ] limiting cases γ/2 ω B k + = ω Aω B 2πγ exp U /k B T γ/2 ω B k + = ω A 2π exp U /k B T transmission coefficient k + = κk TST + high friction low friction