Molecular Dynamics. Park City June 2005 Tully

Transcription

1 Molecular Dynamics

2 John Lance Natasa Vinod Xiaosong Dufie Priya Sharani Hongzhi Group: August, 2004

3 Prelude: Classical Mechanics Newton s equations: F = ma = mq = p Force is the gradient of the potential: F = Vq ( ) q mq = p = V ( q) q Newton s Equations

4 Prelude: Classical Mechanics Hamilton s equations: Define total energy = Hamiltonian, H: q = H ( p, q) p p = H ( q, p) q Hamilton s Equations if H 2 ( pq, ) = T ( p) + Vq ( ) = p /2 m+ Vq ( ) q = p/ m p = V( q) q

5 Prelude: Classical Mechanics Lagrangian Mechanics: Define Lagrangian L: L( qq, ) = T ( q ) Vq ( ) d L L = dt q q Euler-Lagrange Equations if L( qq, ) = T ( q ) Vq ( ) = mq Vq ( ) mq = p = V ( q) q

6 Prelude: Classical Mechanics Hamilton-Jacobi Equation: Define Hamilton s Principle Function S: S = pdq = classical action integral S t = -H q, S q Hamilton-Jacobi Equations S S Vq ( ) + H q, = p + = 0 q t q q

7 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

8 I. The Potential Energy Surface Objective: i Ψ = Ψ t ( rr,, t) H ( rr, ) ( rr,, t) r = electrons R = nuclei H = + = + 2M 2m 2M α R r V α Rα el α i e α α ( rr, ) H ( rr ; ) H ( rr ; ) Φ ( rr ; ) = E ( R) Φ ( rr ; ) el j j j Adiabatic (Born-Oppenheimer) Potential Energy Surface

9 I. The Potential Energy Surface Born-Oppenheimer Approximation: ( rr,, t) ( rr ; ) ( R, t) Ψ Φ Ω j * Substitute into TDSE, multiply from left by Φ rr ;, integrate over r: i Ω j t = <Φ j el Φ j >Ω j t t ( R, ) ( r, R) H ( r, R) ( R, ) 2 2 <Φ j(, rr) R Φ j(, rr) Ω j( R) > α α 2M α j j ( ) = E Ω <Φ Φ > 2 2 j( R) j( R, t) [ j( rr, ) j( rr, ) R α 2Mα + <Φ Φ > + <Φ Φ > Ω 2 2 j( rr, ) R j( rr, ) R j( rr, ) R j( rr, ) j( R, t) α α α α ]

10 I. The Potential Energy Surface Born-Oppenheimer Approximation: 2 Ω 2 j ( R, ) = E j( R) Ω j( R, ) [ <Φ j( r, R) Φ j( r, R) > R t α 2Mα i t t 2 2 j( rr, ) R j( rr, ) R j( rr, ) R j( rr, ) j( R, t) + <Φ Φ > + <Φ Φ > Ω α α α ] α <Φ (, rr) Φ (, rr) > = (1) = 0 R j j R α = <Φ (, rr) Φ (, rr) >+< Φ (, rr) Φ (, rr) > j R j R j j α α α i R, t ( R) R, t ( R) 2 Ω 2 j( ) = E j Ω j( ) Rα Ω j t α 2Mα 2 j(, rr) R j(, rr) j( R,) t +<Φ Φ >Ω α???

11 I. The Potential Energy Surface Born-Oppenheimer Approximation: i t t 2 Ω j = Rα Ω j + j Ω j t α 2Mα 2 ( R, ) ( R) E ( R) ( R, ) where E ( R) = <Φ ( r, R) H Φ ( r, R) > j j el j

12 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

13 II. Classical Limit via Bohm Eqs. 2 i Ω j t = R j j t α t + Ω α 2M α ( R 2, ) E ( R ) ( R, ) i Ω j( R, t) = Aj( R, t) exp[ Sj( R, t)] Substitute (2) into (1) and separate real and imaginary parts: (1) (2) i S [ ] 2 R A α = S E ( R) α 2M 2M A j R j j α α α α j j (3) i 1 A = A S A S 2 { 2 2[ ] i[ ] } j R j R j j R j α M α α α α (4)

14 II. Classical Limit via Bohm Eqs. Compare Eq. (3) with Hamilton-Jacobi Equation: S t = -H q, S q where S q = p i S [ ] 2 R A α = S E ( R) α 2M 2M A j R j j α α α the quantum potential α j j (3) 0: i S 1 [ ] 2 = S α 2M E j R j j α α ( R)

15 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

16 III. Adiabatic on-the-fly Dynamics The Hellman Feynman Theorem: d E dr d ( R) = Φ ( R) H ( R) Φ ( R) dr j j el j subject to Φ ( R) Φ ( R) = 1 j j and H ( R) Φ ( R) = E ( R) Φ ( R) el j j j d E dr dh el ( R) ( R) = Φ ( R) Φ ( R) dr j j j d d + Φ j( R) H el( R) Φ j( R) + Φ j( R) H el( R) Φ j( R) dr dr

17 III. Adiabatic on-the-fly Dynamics The Hellman Feynman Theorem: d E dr dh el ( R) ( R) = Φ ( R) Φ ( R) dr j j j d d + Φ j( R) H el( R) Φ j( R) + Φ j( R) H el( R) Φ j( R) dr dr d E dr dh el ( R) ( R) = Φ ( R) Φ ( R) dr j j j d d + E j( R) Φ j( R) Φ j( R) + Φ j( R) Φ j( R) dr dr d = j( R) j( R) 0 dr Φ Φ =

18 III. Adiabatic on-the-fly Dynamics QM/MM Approach: empirical force field ab initio Brownian dynamics dielectric continuum generalized Langevin

19 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

20 IV. Car-Parrinello Dynamics Euler-Lagrange equations: ddl dt dq L = q Classical Lagrangian: L T V 1 M R 2 E ( R) Car-Parrinello Lagrangian: L = = α 2 α j M R = E ( R)/ R 1 2 α α j α = M R <Ψ H Ψ > α j el j + µ < ϕ ϕ >+ λ [ < ϕ ϕ > δ ] n M R = <Ψ H Ψ > / R α α j el j α α n n n nm n m nm nm µ ϕ = δ <Ψ H Ψ > / δϕ + λ ϕ * n n j el j n nm m m

21 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

22 V. Beyond Born-Oppenheimer

23 V. Beyond Born-Oppenheimer ( rr, ) ( rr ; ) ( R) ( rr, ) ( rr ; ) ( R) Ψ Φ Ω Ψ = Φ Ω j j i i i ( ) Substitute into TISE, multiply from left by Φ j rr ;, integrate over r: Mα R Ω j( R) + E j( R) Ω j( R) E Ω j( R) = α 2 α 2 2 Dji( R) Ω i ( R) + d ji( R) R Ωi ( R) α 2 i i j * where nonadiabatic (derivative) couplings are defined by: { } 1 * ( ) = M Φ (, ) Φ (, ) d R rr rr dr ij α i Rα j α { } 1 * 2 ( ) = Φ (, ) Φ (, ) D R M r R r R dr ij α i Rα j α

24 V. Beyond Born-Oppenheimer Potential Energy Curves for the Oxygen Molecule from R. P. Saxon and B. Liu, J. Chem. Phys energy (ev) internuclear distance (A)

25 V. Beyond Born-Oppenheimer Nonadiabatic Transitions at Metal Surfaces: Electron-Hole Pairs E vac conduction band E F Metal

26 V. Beyond Born-Oppenheimer Vibrational Lifetime of CO on Cu(100) v = 1 v = 0 Molecular Dynamics: t ~ 10-3 s. Experiment (A. Harris et al.): t = 2.5 ps.

27 V. Beyond Born-Oppenheimer H 22 (ionic) H 11 (neutral) off-diagonal coupling: H 12

28 V. Beyond Born-Oppenheimer H ( R) H ( R) H ( R) = H12( R) H22( R) E(R) H 11 (R) H 22 (R) The Non-Crossing Rule diabatic potential energy curves H 12 (R) R E ± H ( R) + H ( R) 1 [ 11 ( ) 22 ( )] 4[ 12 ( )] ( R) = ± H R H R + H R

29 V. Beyond Born-Oppenheimer H ( R) H ( R) H ( R) = H12( R) H22( R) The Non-Crossing Rule E(R) 2H 12 diabatic potential energy curves R adiabatic potential energy curves E ± H ( R) + H ( R) 1 [ 11 ( ) 22 ( )] 4[ 12 ( )] ( R) = ± H R H R + H R

30 V. Beyond Born-Oppenheimer The non-crossing rule for more than 1 degree of freedom: Conical Intersection N degrees of freedom: N-2 dimensional seam E ± H ( R, R ) + H ( R, R ) 1 [ (, ) (, ) ( R, R ) = ± H R R H R R ] + 4[ H ( R, R )]

31 V. Beyond Born-Oppenheimer Thanks to Rick Heller

32 V. Beyond Born-Oppenheimer? The Massey Criterion: E(R) 2H 12? E t E t 2 H 12 distance/velocity R 2 H / ( H H )/ R / R H / ( H H )/ R R ( H H )/ R H12 << 1 adiabatic

33 V. Beyond Born-Oppenheimer? Landau-Zener Approximation E(R) R? Assumptions: 1. H 11 and H 22 linear 2. H 12 constant 3. Velocity constant P nonad 2 2π H 12 exp R ( H11 H22)/ R

34 V. Beyond Born-Oppenheimer Classical motion induces electronic transitions Quantum state determines classical forces Quantum Classical Feedback: Self-Consistency

35 V. Beyond Born-Oppenheimer TWO GENERAL MIXED QUANTUM-CLASSICAL APPROACHES FOR INCLUDING FEEDBACK V b V b V a V a Ehrenfest (SCF) Surface-Hopping

36 V. Beyond Born-Oppenheimer Ψ(,) rt i = H elψ(,) r t R(t) t Ψ (,) rt = c() t Φ (; rr) i i i (adiabatic states). i dc dt = V c R <Φ (; r R) Φ (; r R) > c j jj j j R i i i

37 V. Beyond Born-Oppenheimer Ψ(,) rt i = H elψ(,) r t R(t) t Classical path must respond self-consistently to quantum transitions: quantum back-reaction Ehrenfest and Surface Hopping differ only in how classical path is defined

38 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

39 VI. Ehrenfest Dynamics i Ψ = Ψ t ( rr,, t) H ( rr, ) ( rr,, t) (1) Self-consistent Field Approximation (fully quantum): t i Ψ ( rr,, t) = Ξ( r, t) Ω( R, t) exp Er ( t ') dt' ( ) Substituting (2) into (1), multiplying on the left by Ω R,t and integrating over R gives the SCF equation for the electronic wave function Ξ r,t : ( r, t) 2 Ξ 2 i = i Ξ t + VrR Ξ t t 2m e i ( ) ( r, ) ( r, R) ( r, ) (2) (3) rr * (, ) = Ω (, ) (, ) Ω(, ) V rr Rt V rr Rt dr where (4) rr

40 VI. Ehrenfest Dynamics ( ) Substituting (2) into (1), multiplying on the left by Ξ r,t and integrating over r gives the equivalent SCF equation for the nuclear wave function Ω R,t : ( R, t) 2 Ω 1 2 i = Mα R Ω t α t 2 ( R, ) ( r, t) H ( r, R) ( r, t) dr ( R, t) * + Ξ el Ξ Ω The classical (Ehrenfest) limit requires 2 steps: ( ) 1. Replace Ω R,t with a delta function in Eq. (4) α ( ) 2. Take the classical limit of Eq. (5) (eg. using the Bohm formulation as above). Thus, the potential energy function governing the nuclei becomes * Ξ ( r, t) H el ( r, R) Ξ( r, t) dr instead of the adiabatic energy E j (R). (5)

41 VI. Ehrenfest Dynamics Ψ(,) rt i = H elψ(,) r t R(t) t.. M Rt () = <Ψ() t H Ψ () t > R el V b V a

42 VI. Ehrenfest Dynamics.. M Rt () = <Ψ() t H Ψ () t > R el? Problem: single configuration average path

43 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

44 VII. Surface Hopping Ψ(,) rt i = H elψ(,) r t R(t) t V b V a

45 VII. Surface Hopping Multi-Configuration Wave Function: Ψ (, rr,) t = Φ (, rr) Ω ( R,) t j Substitute into Schrodinger Eq and take classical limit: j j Surface Hopping However, a rigorous classical limit has not been achieved!

46 VII. Surface Hopping One Approach: Multi-Configuration Bohm Equations: Ψ (, rr,) t = Φ (, rr) Ω ( R,) t j Ω (,) (,)exp i j R t = Aj R t S j( R,) t j ( ) 2 R S j = RSj E j ( R ) 2M 2M Aj j A j small h 1 2 j = ( ) ( ) 2 M R j j R S S E motion on potential energy surface j

47 VII. Surface Hopping 1 2 An = RAniR An RSn 2M m i Am <Φn RΦ m > ir exp ( Sm Sn) Surface Hopping: Evaluate all quantities along a single path Sum over many stochastic paths

48 VII. Surface Hopping Multi-Configuration Theory: Surface Hopping Ψ() t i = H elψ() t R(t) t 1].. MRt () = R Ek, i.e., motion on single p.e.s. 2] Stochastic hops between states so that probability = c k 2 3] Apply instantaneous Pechukas Force to conserve energy 4] Fewest Switches : achieve [2] with fewest possible hops:

49 VII. Surface Hopping c 2 2 = 0.8 c 2 2 = 0.7 8? t k t k+1 Stochastic Fewest Switches algorithm (2-state):

50 VII. Surface Hopping c 2 2 = 0.8 c 2 2 = t k t k+1 Stochastic Fewest Switches algorithm (2-state):

51 VII. Surface Hopping c 2 2 = 0.8 c 2 2 = t k t k+1 Stochastic Fewest Switches algorithm (2-state): 2 2 c2( k) c2( k+ 1), c ( k) > c ( k+ 1) 2 P2 1 = c2( k) 0, c ( k) c ( k+ 1)

52 VII. Surface Hopping

53 VII. Surface Hopping Probability of upper state Surface Hopping Exact QM momentum (a.u.)

54 VII. Surface Hopping

55 VII. Surface Hopping 1.0 Probability of upper state Surface Hopping Exact QM log e kinetic energy (a.u.)

56 VII. Surface Hopping Ψ(,) rt i = H elψ(,) r t R(t) t V b Nancy Makri Path V a Jack Simons Path

57 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 1] Trajectories are independent Trajectories should talk to each other quantum wave packet surface hopping Fundamental approximation, but required to make practical

58 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 2] Too drastic: hops require sudden change of velocity Consider swarm of trajectories trajectories hop stochastically at different times gradual evolution of flux

59 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 3] Trajectories should evolve on some effective potential, not on a single adiabatic potential energy surface Consider swarm of trajectories trajectories hop stochastically at different times: t 1 t 2 sources and sinks

60 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 4] Not invariant to representation adiabatic representation diabatic representation The natural representation for surface hopping is adiabatic

61 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 5] Quantum Mechanical Coherence Neglected uses probabilities, not amplitudes FALSE Ψ () t = c () t ϕ ( R) i i i i. dc dt = V c R < ϕ ϕ > c j jj j j R i i i Exact QM Surface Hopping

62 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 6] Forbidden Hops (or frustrated hops) Hopping algorithm calls for a hop but there is insufficient kinetic energy? E total probability on state k = c k 2

63 VII. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 7] Detailed Balance? What are the populations of the quantum states at equilibrium?

64 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

65 VIII. Equilibrium in Mixed Quantum-Classical Dynamics Detailed Balance: N 1 P 1 2 = N 2 P 2 1 Equilibrium Long Timescales Multiple Transitions Relaxation Processes hν Infrequent events e.g., nonradiative transition vs. reaction on excited state

66 VIII. Equilibrium in Mixed Quantum-Classical Dynamics Detailed Balance: N 1 P 1 2 = N 2 P 2 1 Ψ(,) rt i = H elψ(,) r t t R(t). i dc dt = V c R <Φ (; r R) Φ (; r R) > c j jj j j R i i i time reversible P 1 2 = P 2 1 c 1 2 = c 2 2 probabilities of each quantum state are equal: infinite temperature

67 VIII. Equilibrium in Mixed Quantum-Classical Dynamics In theories in which the reservoir is treated classically and its effects on the system described in terms of random functions instead of noncommuting operators, it follows that W mn = W nm. This is a serious shortcoming of all semiclassical theories of relaxation. K. Blum, Density Matrix Theory and Applications, 2 nd Ed., (Plenum, NY, 1996). However This is not true for either Ehrenfest or Surface Hopping

68 VIII. Equilibrium in Mixed Quantum-Classical Dynamics Ehrenfest (SCF) ρ( E ) H( E E ) exp( βe ) de QM TOT QM TOT TOT 0 E E ρ( E ) de ρ( E ) de = QM QM QM QM QM QM exp( β ) E QM = 2 c β 1 exp( β 2 ) 1 exp( β ) = β 1 exp( β ) Boltzmann: E QM = exp( β ) 1+ exp( β ) c 2 2 = exp( β ) 1 + exp( β )

69 VIII. Equilibrium in Mixed Quantum-Classical Dynamics log e (population) 2-state system E = 34.6 kj/mole Excited State Population vs. Inverse Temperature Boltzmann Ehrenfest kτ - exp(- Ε/kΤ) Ε 1-exp(- Ε/kΤ) /T (K)

70 VIII. Equilibrium in Mixed Quantum-Classical Dynamics What about surface hopping? V b. i dc dt = V c R <Φ (; r R) Φ (; r R) > c j jj j j R i i i V a time reversible P 1 2 = P 2 1 c 1 2 = c 2 2 probabilities of each quantum state are equal: infinite temperature looks bad

71 VIII. Equilibrium in Mixed Quantum-Classical Dynamics What about surface hopping? V b. i dc dt = V c R <Φ (; r R) Φ (; r R) > c j jj j j R i i i V a time reversible P 1 2 = P 2 1 c 1 2 = c 2 2 but surface hopping probabilities = c k 2 frustrated hops

72 VIII. Equilibrium in Mixed Quantum-Classical Dynamics ENERGY ENERGY 1 2 X frustrated hop! More configurations with low energy

73 VIII. Equilibrium in Mixed Quantum-Classical Dynamics V b E tot??? frustrated hop V a P 1 2 = exp( - E / kt ) P 2 1 T quantum correct

74 VIII. Equilibrium in Mixed Quantum-Classical Dynamics P hop = - c k 2 / c k 2 = fractional decrease of state k population = 2 Re[ c c ]/ c.. * 2 k k k i dc dt = V c R <Φ (; r R) Φ (; r R) > c j jj j j R i i i Adiabatic representation Phop v N P v ρ( v ) dv v exp( mv / kt) dv exp( E / kt) 1 N 2 1 P1 2 = N 2 P2 1 = exp[ ( E2 E1) / kt] N 1

75 VIII. Equilibrium in Mixed Quantum-Classical Dynamics 2-State System E = 34.6 kj/mole log e (population) Excited State Population vs. Inverse Temperature Boltzmann kτ - exp(- Ε/kΤ) Ε 1-exp(- Ε/kΤ) Ehrenfest Surface Hopping /T (K)

76 VIII. Equilibrium in Mixed Quantum-Classical Dynamics Many Quantum States [classical bath] --- C --- C --- C --- C --- C --- C --- C --- C --- H Random force and friction (5000 o K) classical (Morse) quantum (Morse) Ehrenfest (SCF) Surface Hopping V=0 Priya Parandekar V=1 V=2 V=3

77 Molecular Dynamics I. The Potential Energy Surface II. The Classical Limit via the Bohm Equations III. Adiabatic on-the-fly Dynamics IV. Car-Parrinello Dynamics V. Beyond Born Oppenheimer VI. Ehrenfest Dynamics VII. Surface Hopping VIII. Equilibrium in Mixed Quantum-Classical Dynamics IX. Mixed Quantum-Classical Nuclear Motion

78 XI. Mixed Quantum-Classical Nuclear Motion Tenets of Conventional Molecular Dynamics 1. The Born-Oppenheimer Multiple Electronic Approximation States, Metals, 2. Classical Mechanical Nuclear Motion Zero Point Motion, Quantized Energy Levels, Tunneling

79 XI. Mixed Quantum-Classical Nuclear Motion Quantum Effects: Zero-Point Energy Quantized Energy Levels Tunneling

80 XI. Mixed Quantum-Classical Nuclear Motion Ultimate Solution: Treat All Electrons and Nuclei by Quantum Mechanics Problem: Scaling with Size is Prohibitive

81 XI. Mixed Quantum-Classical Nuclear Motion AN ALTERNATIVE STRATEGY: MIXED QUANTUM CLASSICAL DYNAMICS quantum dynamics classical dynamics

82 XI. Mixed Quantum-Classical Nuclear Motion AN ALTERNATIVE STRATEGY: MIXED QUANTUM CLASSICAL DYNAMICS Treat crucial electronic or nuclear degrees of freedom by quantum mechanics, and the remaining nuclei by classical mechanics. self-consistency quantum back-reaction on classical particles

83 XI. Mixed Quantum-Classical Nuclear Motion PROTON TRANSFER IN SOLUTION covalent reactant: self consistency! ionic product:

84 XI. Mixed Quantum-Classical Nuclear Motion ADIABATIC vs. NON-ADIABATIC

85 XI. Mixed Quantum-Classical Nuclear Motion ADIABATIC vs. NON-ADIABATIC

86 XI. Mixed Quantum-Classical Nuclear Motion R AB A----H----B r

87 XI. Mixed Quantum-Classical Nuclear Motion Proton and Hydride Transfer in Enzymes: Sharon Hammes-Schiffer Penn State University liver alcohol dehydrogenase (LADH)

88 XI. Mixed Quantum-Classical Nuclear Motion liver alcohol dehydrogenase Sharon Hammes-Schiffer