Molecular Dynamics. Park City June 2005 Tully

Molecular Dynamics

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

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

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

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 ( ) 1 2 2 mq = p = V ( q) q

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

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

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 2 2 2 2 2 2 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

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 ]

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???

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

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 2 1 [ ] 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)

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 2 1 [ ] 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)

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

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 =

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

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 > 1 2 2 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

V. Beyond Born-Oppenheimer

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: 2 1 2 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

V. Beyond Born-Oppenheimer Potential Energy Curves for the Oxygen Molecule from R. P. Saxon and B. Liu, J. Chem. Phys. 1977 energy (ev) 15 10 5 0 1 2 3 internuclear distance (A)

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

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.

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

V. Beyond Born-Oppenheimer H ( R) H ( R) H ( R) 11 12 = 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 ( )] 2 2 11 22 2 2 ( R) = ± H R H R + H R

V. Beyond Born-Oppenheimer H ( R) H ( R) H ( R) 11 12 = 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 ( )] 2 2 11 22 2 2 ( R) = ± H R H R + H R

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 [ (, ) (, ) 2 2 11 1 2 22 1 2 2 2 ( R, R ) = ± H R R H R R ] + 4[ H ( R, R )] 1 2 11 1 2 22 1 2 12 1 2

V. Beyond Born-Oppenheimer Thanks to Rick Heller

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 12 11 22 2 H / ( H H )/ R 12 11 22 R ( H H )/ R 11 22 2 4H12 << 1 adiabatic

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

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

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

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

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

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

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)

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

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

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

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!

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 2 1 ( ) 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

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

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:

VII. Surface Hopping c 2 2 = 0.8 c 2 2 = 0.7 8? 7 2 3 t k t k+1 Stochastic Fewest Switches algorithm (2-state):

VII. Surface Hopping c 2 2 = 0.8 c 2 2 = 0.7 8 7 2 3 t k t k+1 Stochastic Fewest Switches algorithm (2-state):

VII. Surface Hopping c 2 2 = 0.8 c 2 2 = 0.7 8 7 2 3 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) 2 2 2 2 2 2 2 2

VII. Surface Hopping

VII. Surface Hopping 0.8 0.7 Probability of upper state 0.6 0.5 0.4 0.3 0.2 Surface Hopping Exact QM 0.1 0 5 10 15 20 25 30 momentum (a.u.)

VII. Surface Hopping

VII. Surface Hopping 1.0 Probability of upper state 0.8 0.6 0.4 0.2 0.0 Surface Hopping Exact QM -4-3 -2-1 0 1 log e kinetic energy (a.u.)

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

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

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

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

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

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

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

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

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

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

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

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 0 0 1 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 )

VIII. Equilibrium in Mixed Quantum-Classical Dynamics log e (population) 2-state system E = 34.6 kj/mole 0-2 -4-6 -8-10 Excited State Population vs. Inverse Temperature Boltzmann Ehrenfest kτ - exp(- Ε/kΤ) Ε 1-exp(- Ε/kΤ) 0 0.001 0.002 1/T (K)

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

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

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

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

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 1 2 1 2 1 1 1 1 2 1 1 exp( E / kt) 1 N 2 1 P1 2 = N 2 P2 1 = exp[ ( E2 E1) / kt] N 1

VIII. Equilibrium in Mixed Quantum-Classical Dynamics 2-State System E = 34.6 kj/mole log e (population) 0-2 -4-6 -8 Excited State Population vs. Inverse Temperature Boltzmann kτ - exp(- Ε/kΤ) Ε 1-exp(- Ε/kΤ) Ehrenfest Surface Hopping -10 0 0.001 0.002 1/T (K)

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

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

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

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

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

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

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

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

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

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

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