IV. Classical Molecular Dynamics

Transcription

1 IV. Classical Molecular Dynamics Basic Assumptions: 1. Born-Oppenheimer Approximation 2. Classical mechanical nuclear motion Unavoidable Additional Approximations: 1. Approximate potential energy surface 2. Incomplete sampling 3. (often) Extrapolate to longer timescales 4. Too few atoms 5. Continuum solvent and other shortcuts

2 V. Adiabatic on-the-fly Dynamics What about the potential energy surface? i.e., the forces? empirical (sometimes called classical - bad terminology) semi-empirical (Huckel, valence-bond, PPP-Rossky ) analytic fit to ab initio (HF, DFT, MC-SCF, CCSD, ) on-the-fly ab initio (HF, DFT,, )

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

4 V. Adiabatic on-the-fly Dynamics What about the potential energy surface? i.e., the forces? empirical (sometimes called classical - bad terminology) semi-empirical (Huckel, valence-bond, PPP-Rossky ) analytic fit to ab initio (HF, DFT, MC-SCF, CCSD, ) on-the-fly ab initio (HF, DFT,, )

5 Chemical Dynamics I. Quantum Dynamics II. Semiclassical Dynamics aside: tutorial on classical mechanics III. The Classical Limit via the Bohm Equations IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics VII. Infrequent Events aside: transition state theory and re-crossing VIII. Beyond Born Oppenheimer IX. Ehrenfest Dynamics X. Surface Hopping XI. Dynamics at Metal Surfaces XII. Mixed Quantum-Classical Nuclear Dynamics

6 V. Adiabatic on-the-fly Dynamics The Hellman Feynman Theorem: d E dr d ( R) ( R) H ( R) ( R) dr j j el j = force subject to ( R) ( R) 1 j j and H ( R) ( R) E ( R) ( R) el j j j d E dr dh el ( R) ( R) ( R) ( R) dr j j j d d j( R) H el( R) j( R) j( R) H el( R) j( R) dr dr

7 V. Adiabatic on-the-fly Dynamics The Hellman Feynman Theorem: d E dr dh el ( R) ( R) ( R) ( R) dr j j j d d j( R) H el( R) j( R) j( R) H el( R) j( R) dr dr d E dr dh el ( R) ( R) ( R) ( R) dr j j j d d E j( R) j( R) j( R) j( R) j( R) dr dr Pulay corrections d j( R) j( R) 0 dr

8 Chemical Dynamics I. Quantum Dynamics II. Semiclassical Dynamics aside: tutorial on classical mechanics III. The Classical Limit via the Bohm Equations IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics VII. Infrequent Events aside: transition state theory and re-crossing VIII. Beyond Born Oppenheimer IX. Ehrenfest Dynamics X. Surface Hopping XI. Dynamics at Metal Surfaces XII. Mixed Quantum-Classical Nuclear Dynamics

9 VI. Car-Parrinello Dynamics Euler-Lagrange equations: ddl L dt dq q Classical Lagrangian: L T V 1 M R 2 E R 2 j M R E ( R)/ R j Car-Parrinello Lagrangian: fictitious electronic dynamics L M R H j el j [ ] n n n n nm n m nm nm Kohn-Sham orbital Roberto Car, Michele Parrinello

10 VI. Car-Parrinello Dynamics Student problem: Lagrange (undetermined) parameter problem find the minimum of the function subject to x y 2 f ( xy, ) xy 2 2, using the method of Lagrange parameters define hxy (, ) f( xy, ) ( x y2) and minimize wrt x, y and

11 VI. Car-Parrinello Dynamics Euler-Lagrange equations: ddl L dt dq q Classical Lagrangian: L T V 1 M R 2 E R 2 j M R E ( R)/ R j Car-Parrinello Lagrangian: L M R H j el j [ ] n n n n nm n m nm nm M R H / R j el j H / * n n j el j n nm m m Roberto Car, Michele Parrinello

12 IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics CAUTION! exact PES + exact quantum dynamics exact exact PES + classical dynamics inaccurate??? empirical PES + classical dynamics maybe accurate??? is first-principles synonymous with less accurate?

13 Chemical Dynamics I. Quantum Dynamics II. Semiclassical Dynamics aside: tutorial on classical mechanics III. The Classical Limit via the Bohm Equations IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics VII. Infrequent Events aside: transition state theory and re-crossing VIII. Beyond Born Oppenheimer IX. Ehrenfest Dynamics X. Surface Hopping XI. Dynamics at Metal Surfaces XII. Mixed Quantum-Classical Nuclear Dynamics

14 VII. Infrequent Events Molecular Dynamics Time Step: ~ s Feasible number of integration steps: ~ Maximum Time for Direct MD Integration: ~ 10-6 s Moreover, need to sample multiple trajectories, different conditions, etc. Examples: High activation energy processes Slow relaxation times (timescale mismatch, glasses, ) Concerted movement of many atoms (protein folding) Infrequent Event

15 VII. Infrequent Events Many methods to enhance sampling for equilibrium simulations. Non-Boltzmann sampling, e.g., umbrella sampling energy U(x) V(x)+U(x) Un-bias by weighting each point by exp[+u(x)/kt] V(x) But what about dynamics? reaction coordinate

16 VII. Infrequent Events Reaction Rate Theory (Classical) A S 1. Define reactant and product regions and equilibrium reaction rate R 1 S < S B 2. Define reaction variable S and dividing surface S R 2 S > S 3. Compute one-way flux through dividing plane: k forward only 0 G ( p ) s p s m dp s fraction of molecules at S = S with momentum p s velocity at S

17 VII. Infrequent Events G ( p ) s define:... exp[ H ( s, p, q, p ) / kt ] ( s s ) ds dq dp s... exp[ H ( s, ps, q, pq) / kt ] ds dps dq dpq s q q s Q0 exp( E0 / kt) M h... exp[ H ( s, ps, q, pq) / kt] ds dps dq dp q Planck s constant: conventional quantum-classical correspondence E E 0 s a

18 VII. Infrequent Events G ( p ) s define:... exp[ H ( s, p, q, p ) / kt ] ( s s ) ds dq dp s... exp[ H ( s, ps, q, pq) / kt ] ds dps dq dpq s q q s Q0 exp( E0 / kt) M h... exp[ H ( s, ps, q, pq) / kt] ds dps dq dp q Planck s constant: conventional quantum-classical correspondence student problem: simple harmonic oscillator: V( x) K x m x show that at high temperatures, the quantum and classical partition functions are equal if the classical partition function is defined by 1 2 Qclassical exp V( x) / ktexp p / 2mkT dxdp h

19 VII. Infrequent Events G ( p ) s... exp[ H ( s, p, q, p ) / kt ] ( s s ) ds dq dp s s q q... exp[ H ( s, p, q, p ) / kt ] ds dp dq dp s q s q define: and s M Q0 exp( E0 / kt) h... exp[ H ( s, p, q, p ) / kt] ds dp dq dp s q s q ( 1) exp( / ) M Q E kt h... exp[ H ( s, q, pq) / kt] dq dpq E where H ( s, q, p ) H( s, p, q, p ) p /2m 2 q s q s E 0 s a

20 VII. Infrequent Events G p p Q E E kt 2 s ( s ) exp exp[ ( 0) / ] 2mkT hq0 k p p Q ( E E ) 2 s s 0 exp exp 0 m 2mkT hq0 kt dp s k kt Q ( E E ) 0 exp h Q0 kt transition state theory Activated Complex Theory equilibration? Lake Eyring Henry Eyring

21 VII. Infrequent Events Transition State Theory is an upper limit to the true reaction rate Variational Transition State Theory A VTST: Choose location of dividing surface to minimize transition state theory rate R 1 S B S < S But still not the true rate S > S R 2 Recrossing correction

22 VII. Infrequent Events Recrossing correction: rate k f k 1. f = fraction of reactive trajectories, f < 1 A S 2. TST rate depends on location of dividing surface S. R 1 B 3. True rate does not depend on location of dividing surface S. 4. k k R 2

23 Chemical Dynamics I. Quantum Dynamics II. Semiclassical Dynamics aside: tutorial on classical mechanics III. The Classical Limit via the Bohm Equations IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics VII. Infrequent Events aside: transition state theory and re-crossing VIII. Beyond Born Oppenheimer IX. Ehrenfest Dynamics X. Surface Hopping XI. Dynamics at Metal Surfaces XII. Mixed Quantum-Classical Nuclear Dynamics

24 VIII. Beyond Born-Oppenheimer Ahmed Zewail

25 VIII. Beyond Born-Oppenheimer rr, rr ; R Born-Oppenheimer i i

26 VIII. Beyond Born-Oppenheimer rr, rr ; R i Beyond Born-Oppenheimer i i

27 VIII. Beyond Born-Oppenheimer Substitute into TISE, multiply from left by j rr ;, integrate over r: M R jre jr jr E jr D jir i R d jirr i R 2 i i j student problem: show that d ij (R) is anti-hermitian and its diagonal elements are zero where nonadiabatic (derivative) couplings are defined by: 1 * M,, d R rr rr dr ij i R j 1 * 2,, D R M r R r R dr ij i R j rr, rr ; R i i * i

28 VIII. 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)

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

30 VIII. 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.

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

32 VIII. Beyond Born-Oppenheimer The Non-Crossing Rule H R H ( R) H ( R) H12( R) H22( R) E(R) H 11 (R) H 22 (R) diabatic potential energy curves H 12 (R) R H. C. Longuet-Higgins E H ( R) H ( R) 1 [ 11 ( ) 22 ( )] 4[ 12 ( )] R H R H R H R

33 VIII. 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

34 VIII. 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 ) R, R [ H ( R, R ) H ( R, R )] 4[ H ( R, R )]

35 VIII. 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 H. S. W. Massey

36 VIII. 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 2 H 12 exp R ( H11 H22)/ R L. D. Landau C. M. Zener

37 VIII. Beyond Born-Oppenheimer How do we simulate dynamics? Classical motion induces electronic transitions Quantum state determines classical forces Quantum Classical Feedback: Self-Consistency

38 VIII. Beyond Born-Oppenheimer rr, rr ; R Substitute into TISE, multiply from left by j rr ;, integrate over r: M R jre jr jr E jr DjiR i R d jirr i R 2 i i j i i * i Wave packet methods: Initial wave function: Gaussian wave packet 1/ (,0) exp( 2 0 ) exp[ ( 0) / ] i x ik x x x a a j ( x,0) 0, j i

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

40 VIII. Beyond Born-Oppenheimer (,) rt i H el(,) r t R(t) t (,) rt c() t (; rr) i i i (adiabatic states). i dc dt V c R (; r R) (; r R) c j jj j j R i i i

41 VIII. Beyond Born-Oppenheimer i (,) rt t H el (,) r t R(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

42 Chemical Dynamics I. Quantum Dynamics II. Semiclassical Dynamics aside: tutorial on classical mechanics III. The Classical Limit via the Bohm Equations IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics VII. Infrequent Events aside: transition state theory and re-crossing VIII. Beyond Born Oppenheimer IX. Ehrenfest Dynamics X. Surface Hopping XI. Dynamics at Metal Surfaces XII. Mixed Quantum-Classical Nuclear Dynamics Paul Ehrenfest

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

44 IX. Ehrenfest Dynamics Substituting (2) into (1), multiplying on the left by r,t and integrating over r gives the equivalent SCF equation for the nuclear wave function R,t : 2 R, t 1 2 i M R R, t t 2 r, th r, R r, tdr R, t * el (5) Thus, the potential energy function governing the nuclei becomes, th,, * r el r R r t dr instead of the adiabatic energy E j (R). The Ehrenfest method consists of taking classical limit of Eq. 5, i.e., running classical trajectories subject to this potential function

45 IX. Ehrenfest Dynamics i (,) rt t H el (,) r t R(t).. M Rt () () t H () t R el V b V a

46 IX. Ehrenfest Dynamics.. M Rt () () t H () t R el? Problem: single configuration average path

47 Chemical Dynamics I. Quantum Dynamics II. Semiclassical Dynamics aside: tutorial on classical mechanics III. The Classical Limit via the Bohm Equations IV. Classical Molecular Dynamics V. Adiabatic on-the-fly Dynamics VI. Car-Parrinello Dynamics VII. Infrequent Events aside: transition state theory and re-crossing VIII. Beyond Born Oppenheimer IX. Ehrenfest Dynamics X. Surface Hopping XI. Dynamics at Metal Surfaces XII. Mixed Quantum-Classical Nuclear Dynamics

48 X. Surface Hopping i (,) rt t H el (,) r t R(t) V b V a

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

50 X. Surface Hopping One Approach: Multi-Configuration Bohm Equations: (, rr,) t (, rr) ( R,) t j ( R,) t A ( R,)exp t i S ( R,) t j j j j j S 2 2 R A 2M 2M A 1 ( ) 2 j RSj E j ( R ) j j small h S S E 1 2 j ( ) ( ) 2 M R j j R motion on potential energy surface j

51 X. Surface Hopping 1 2 An RAnR AnRSn 2M m i Am nrm R exp ( Sm Sn) Surface Hopping: Evaluate all quantities along a single path Sum over many stochastic paths

52 X. Surface Hopping i Multi-Configuration Theory: Surface Hopping (,) rt H el(,) r t t R(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:

53 X. 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( k1), c ( k) c ( k1) 2 P2 1 c2( k) 0, c ( k) c ( k1)

54 X. Surface Hopping

55 X. Surface Hopping Probability of upper state Surface Hopping Exact QM momentum (a.u.)

56 X. Surface Hopping

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

58 X. 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

59 X. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 2] 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

60 X. Surface Hopping SHORTCOMINGS OF SURFACE HOPPING 3] Detailed Balance? What are the populations of the quantum states at equilibrium?

61 X. Surface Hopping Detailed Balance: N 1 P 12 = N 2 P 21 Equilibrium Long Timescales Multiple Transitions Relaxation Processes h Infrequent events e.g., nonradiative transition vs. reaction on excited state

62 X. Surface Hopping 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

63 X. Surface Hopping 2-State System E = 34.6 kj/mole log e (population) Excited State Population vs. Inverse Temperature Boltzmann kτ - exp(-δε/kτ) ΔΕ 1-exp(-ΔΕ/kΤ) Ehrenfest Surface Hopping /T (K)

64 X. Surface Hopping 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 V=1 Priya Parandekar V=2 V=3