Ab Initio Molecular Dynamic

Transcription

1 Ab Initio Molecular Dynamic Paul Fleurat-Lessard AtoSim Master / RFCT Ab Initio Molecular Dynamic p.1/67

2 Summary I. Introduction II. Classical Molecular Dynamic III. Ab Initio Molecular Dynamic (AIMD) 1 - Which one? 2 - Car-Parinello! IV. Application : Free Energy Calculations V. Conclusions Ab Initio Molecular Dynamic p.2/67

3 Introduction Goal : Compute statistical averages, as in Monte Carlo. Sample phase space Macroscopic properties Tool : Time evolution of the system. Ab Initio Molecular Dynamic p.3/67

4 Microscopic becomes Macroscopic Trajectories sampling phase space A macro = A = A exp( βe)d ri d p i exp( βe)d ri d p i Ab Initio Molecular Dynamic p.4/67

5 Microscopic becomes Macroscopic Trajectories sampling phase space Ergodic hypothesis verified : A macro = A = lim T AT 1 = lim T T T 0 A(t)dt Ab Initio Molecular Dynamic p.4/67

6 Microscopic becomes Macroscopic Trajectories sampling phase space Ergodic hypothesis verified : A macro = A = lim T AT 1 = lim T T T 0 A(t)dt True only if : System at equilibrium Simulation time T much larger than caracteristic times of the system. Ab Initio Molecular Dynamic p.4/67

7 Short History of MD Simulations using balls (hard and soft) Computer Simulations Super-computer MANIAC (Los Alamos in 1952) Monte Carlo : Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) MD Hard Sphere : Alder and Wainwright (1957) MD with actual movement equations : Rahman (Argonne 1964) Ab Initio Molecular Dynamic Ehrenfest 1927 : First approach. C. Leforestier 1978 : Born-Oppenheimer Car and Parrinello (Sissa 1985) Ab Initio Molecular Dynamic p.5/67

8 Computer simulations Four steps : Initialization Propagation Equilibration Analysis Propagation : All schemes based on numerical solution m i r i = i V ({ r i }) Which V? Empirical : classical force fields Quantum : semi-empirical, ab initio and DFT How to obtain V? Ab Initio Molecular Dynamic p.6/67

9 Computer simulations Four steps : Initialization Propagation Equilibration Analysis Propagation : All schemes based on numerical solution m i r i = i V ({ r i }) Which V? How to obtain V? V pre-calculated and tabulated : limited to few degrees of freedom, allows quantum nuclei ; V calculated on the fly : mostly classical nuclei Ab Initio Molecular Dynamic p.6/67

10 Time scale V Simulation duration (ps) Atoms Force Fields Semi-empirical DFT, HF post-hf 1 10 Ab Initio Molecular Dynamic p.7/67

11 Classical Molecular Dynamic Ab Initio Molecular Dynamic p.8/67

12 Equations of motion Many formalism : Newton Lagrange Hamilton m i r i = i V ({ r i }) Ab Initio Molecular Dynamic p.9/67

13 Equations of motion Many formalism : Newton Lagrange Lagrangian for a system with n degrees of freedom L({ r i },{ ri }) = K({ r i },{ ri }) V ({ r i }) Equation of motion : Euler-Lagrange d dt L L r i = 0 r i Impulsion p i = L, Force F i = L r i r i Newton is found again : p i = F i Ab Initio Molecular Dynamic p.9/67

14 Equations of motion Many formalism : Newton Lagrange Hamilton Impulsion p i used instead of velocity Hamiltonian H({ r i }, { p i }) = 3n i=1 q i p i L r i = K({ r i }, { p i }) + V ({ r i }) Ab Initio Molecular Dynamic p.9/67

15 Equations of motion Many formalism : Newton Lagrange Hamilton Hamiltonian H({ r i }, { p i }) = K({ r i }, { p i }) + V ({ r i }) Equations of motion r i = H p i p i = H r i Ab Initio Molecular Dynamic p.9/67

16 Equations of motion Many formalism : Newton Lagrange Hamilton Equations of motion q i = H p i p i = H q i H is constant along the trajectory, interpreted as the total energy dh dt = [ H r i + H ] p i r i i p i Ab Initio Molecular Dynamic p.9/67

17 Equations of motion Many formalism : Newton Lagrange Hamilton Equations of motion q i = H p i p i = H q i H is constant along the trajectory, interpreted as the total energy dh dt = [ H H H ] H = 0 r i i p i p i q i Ab Initio Molecular Dynamic p.9/67

18 Equations of motion Many formalism : Newton Lagrange Hamilton Conclusion : all equivalent! Ab Initio Molecular Dynamic p.9/67

19 Working ingredients Initialization : { r i }, { p i } at t = 0 { r 0 i } : from cristallographic structure, PDB or by similarity to avoid very repulsive part of the potential. { p 0 i } : most of the time 0, random or taken from Maxwell-Bolztman distribution. How to compute V? Time is discrete, how to choose time step t? As large as possible Smaller than characteristic time of the system Ab Initio Molecular Dynamic p.10/67

20 Empirical potentials Force fields VSEPR extension : Geometries close to a reference : d CC 1, 54 Å, d CH 1, 09 Å, α Ethane : d CC = 1, 536 Å, Propane : d CC = 1, 526 Å, α = 112, 4, Butane : d (e) CC = 1, 533 Å, d(i) CC = 1, 533 Å, α = 112, 8. Energy close the reference energy! E = E 0 + corrections... Ab Initio Molecular Dynamic p.11/67

21 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... Ab Initio Molecular Dynamic p.12/67

22 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... V bonding Bonding energy V bonding = 1 2 i bondings k r,i (r i r 0 i ) 2 Ab Initio Molecular Dynamic p.12/67

23 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... V angle Valence angle (bending) modification V angle = 1 2 k θ,ij (θ ij θij) 0 2 i,j bondings Ab Initio Molecular Dynamic p.12/67

24 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... V torsion Torsion energy (rocking) V torsion = 1 2 [V 1 (1 + cosϕ i )+ i V 2 (1 cos 2ϕ i ) + V 3 (1 + cos 3ϕ i )] Ab Initio Molecular Dynamic p.12/67

25 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... V torsion Torsion energy Energie Eclipsee E(ψ) Energie Butane : V1=1., V2=0., V3=1. Syn Gauche Decalee120 ψ (angle HCCH) ψ (angle CCCC) Anti Ab Initio Molecular Dynamic p.12/67

26 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... V nb Non Bonded interactions Van Der Waals : V V dw (d) = ǫ i,j atomes ( d 0 ij d ij ) 12 2 ( d 0 ij d ij ) 6 Ab Initio Molecular Dynamic p.12/67

27 General form of a Force Field Main contributions : E MM = V bonding + V angle + V torsion + V nb +... V nb Non Bonded interactions electrostatic type : V el = i,j atomes q i q j Dd ij H-bonds Ab Initio Molecular Dynamic p.12/67

28 Popular force fields MM2, MM3 Mainly for organic chemistry. MMFF94 Only for organic chemistre! AMBER, CHARMm Good for biological systems UFF Universal Force Field. Universaly avoided! ESFF Extensible and Systematic Force Field Small benchmark : Ab initio MMFF94 MM3 CHARMm AMBER E 0,4 0,4 0,7 0,8 1,1 d XH - 0,09 0,5 0,1 0,2 Ab Initio Molecular Dynamic p.13/67

29 Propagator A good propagator must : Be quick and allow for large t Time reversible Conserve mechanical energy Compute forces not too frequently Conserve phase-space volume Basic idea : Taylor expansion ri (t + t) = r i (t) + v i (t) t + ai (t) 2 t vi (t + t) = v i (t) + a i (t) t + vi (t) 2 t Ab Initio Molecular Dynamic p.14/67

30 Verlet General expression : ri (t + t) = r i (t) + v i (t) t + ai (t) 2 t ri (t t) = r i (t) ai (t) v i (t) t + 2 t thus r i (t + t) = 2 r i (t) fi (t) r i (t t) + t 2 + O( t 3 ) M i Store position and forces only Accuracy in t 3 Time reversible by construction Velocities indirectly computed (finite differences) : less accurate vi = ( r i (t + t) r i (t t))/(2 t) + O( t 2 Ab Initio Molecular ) Dynamic p.15/67

31 Leap-Frog variant Designed to improve numerical accuracy ri (t + t) = r i (t) + v i (t + t 2 ) t vi (t + t 2 ) = v i (t t 2 ) + a i (t) t vi (t) = ( vi (t + t 2 ) v i (t t 2 )) /2 + O( t 2 ) More stable Ab Initio Molecular Dynamic p.16/67

32 Velocity Verlet Improve velocity accuracy ri (t + t) = r i (t) + v i (t) t + ai (t) 2 t2 vi (t + t) = ai (t) + a i (t + t) v i (t) + t 2 Velocity accuracy in O( t 3 ) Can be seen as a predictor-corrector Ab Initio Molecular Dynamic p.17/67

33 Verlet : Summary Graphical form : Velocity verlet often used because : Simple and efficient, little force computation Accurate in t 3 Time reversible, Symplectic (conserve phase space volume) For realistic time step, Velocity Verlet as good as Gear predictor-corrector Ab Initio Molecular Dynamic p.18/67

34 Constrained Dynamic Why using constraints : to get larger t (frozen d(ch)), prevent or force some evolution Principal : Lagrange multiplier Constraint defined by σ ({ r i }) = 0, For example : Distance : σ ({ r i }) = r i r j d 0 Difference of two distances : σ ({ r i }) = r i r j r k r l d 0 Coordination number : σ ({ r i }) = n i ({ r i }) n 0 with n i ({ r i }) = j i S( r i r j ) et S(r) = (1 + exp(κ(r r c ))) 1. Ab Initio Molecular Dynamic p.19/67

35 Constrained Dynamic Constraint defined by σ ({ r i }) = 0 Extended Lagrangian : L ({ r i }, { p i }) = L({ r i }, { p i }) α λ α σ α ({ r i }) Equations of motion : m i r i = V r i α λ α σ α r i SHAKE : λ α tq σ α ({ r i (t + t)}) = 0 RATTLE : λ α tq σ α ({ r i (t + t)}) = 0 Ab Initio Molecular Dynamic p.19/67

36 Which ensembles? By default, isolated system E conserved, i.e. NVE Initialize { p i } so that E tot = E target then propagate How can we obtain other ensembles? NVT : E fluctuates Q(N, V, T) = 1 N!h 3N NPT : E and V fluctuate 1 (N, P, T) V 0 N!h 3N Z Z Π i d r i d p i exp( βh) = exp ( βf(n, V, T)) Π i d r i d p i exp ( β(h + PV )) = exp( βg(n, P, T)) Ab Initio Molecular Dynamic p.20/67

37 Canonical ensemble We know tha k T (t) = 2 3Nk B k p 2 k 2m k = 3 2 Nk BT leading to p k (t) 2 2m k At equilibrium, Maxwell-Boltzmann distribution Initialize Velocities from MB distribution Or p 0 i = 0, then heating using "inversed annealing" : v i (t) α v i (t), α > 1. Equilibrate : homogenizet, to obtain T (t) = T Ab Initio Molecular Dynamic p.21/67

38 Canonical ensemble We know tha k T (t) = 2 3Nk B k p 2 k 2m k = 3 2 Nk BT leading to p k (t) 2 2m k At equilibrium, Maxwell-Boltzmann distribution Initialize Equilibrate : homogenizet, to obtain T (t) = T Rescaling : when T (t) T > T, then vi (t) v i (t) T T (t) Brutal! Thermostat : coupling with a thermal bath Ab Initio Molecular Dynamic p.21/67

39 Nosé-Hoover Extended System : a supplementary variable that mimics the thermal bath Equations Ab Initio Molecular Dynamic p.22/67

40 Nosé-Hoover Extended System Equations ri = pi m i ξ = 1 Q ( k p 2 k 2m k gk B T ) pi = F i ξ p i = gk B Q (T T) with g number of degrees of freedom, Q mass of the thermostat Ab Initio Molecular Dynamic p.22/67

41 Nosé-Hoover Equations Q regulates the speed of exchanges between the bath and the system Q too small : large not wanted oscillations, slow convergence Q too large : slow exchanges. Q, microcanonical ensemble! Optimum when resonating with the system, ie Q gk BT ω 2 Ab Initio Molecular Dynamic p.22/67

42 Nosé-Hoover Chains Nosé-Hoover not always ergodic, response time too slow Chain of M thermostats : ri = pi m i ξ 1 = 1 Q 1 ( k p 2 k 2m k gk B T ) ξ 1 ξ 2 pi = F i ξ 1pi ξ n = 1 Q n ( Qn 1 ξ 2 n 1 k B T ) ξ n ξ n+1 ξ M = 1 ( QM 1 ξn 1 2 k B T ) Q M Ab Initio Molecular Dynamic p.23/67

43 Test 1D oscillator H = p2 2m + mω2 x 2 2, f(p) = β 2πm e βp2 /2m, f(x) = βmω 2 2π e βω2 x 2 /2 1 thermostat 3 thermostats 4 thermostats Ab Initio Molecular Dynamic p.24/67

44 Andersen or Hybrid Monte Carlo Mix DM and Monte Carlo to simulate chocs with the thermal bath Andersen The velocity of a random particle is withdrawn from the MD distribution ; Ergodic by construction, converges to the NVT associated to the MB distribution ; Many particles can be chosen at once Re-attribution period not so important Re-attribution does not kick wafefunction too far from BO surface Hybrid Monte Carlo : all velocities are changed at once Ab Initio Molecular Dynamic p.25/67

45 Langevin Still stochastic, but white noise added directly in the equation of motion for all particles q = p t m ṗ = V (q) ξ m p + 2ξ β dw(t)/dt with W(t) a gaussian white noise Physically (mathematically?) : the most efficient to thermalize, the fastest to converge but very rare in chemistry, and badly behave in CPMD Ab Initio Molecular Dynamic p.26/67

46 False thermostat : Berendsen Can be seen as a rescaling, with α(t + t) = = [ 1 + t τ [ 1 + ( T T (t + t) 1 t τt (t + t) )] 1/2 (T T (t + t)) T T (t + t) affects α and not α Very used in biochemistry but do not sample the canonical ensemble! Use only for equilibration. ] 1/2 Ab Initio Molecular Dynamic p.27/67

47 Isobaric-Isotherm ensemble This is the closest to chemist experiments Volumebecomes a dynamical variable No stochastic approaches Two families : Isotropic : V change, but not the box shape Hoover, Anderssen Anisotropic :all parameters of the box can change Parrinello-Rahman, Martyna-Tobias-Klein Ab Initio Molecular Dynamic p.28/67

48 Isotropic NPT : Andersen First approach (1980) Mimic the action of a piston on the box : Mass Q Kinetic energy K V = 1 2 Q V 2 Potential energy V V = PV Reduced variables : s = V 1/3 r, ṡ = V 1/3 v Hamiltonian : H V = K + K V + V + V V Ab Initio Molecular Dynamic p.29/67

49 Isotropic NPT : Andersen First approach (1980) Mimic the action of a piston on the box Reduced variables : s = V 1/3 r, ṡ = V 1/3 v Hamiltonian : H V = K + K V + V + V V Equations of motion : f s = mv 2 ṡ V 1/3 3 V V = (P P)/Q with P instantaneous pressure P = 2 3V (K W) et W = 1 2 i j>i r ijfij Internal Virial. Ab Initio Molecular Dynamic p.29/67

50 Isotropic NPT : Andersen First approach (1980) Mimic the action of a piston on the box Reduced variables : s = V 1/3 r, ṡ = V 1/3 v Hamiltonian : H V = K + K V + V + V V Equations of motion : f s = mv 2 ṡ V 1/3 3 V V = (P P)/Q In fact, sampling the isobaric-iso-enthalpic ensemble, not so common. Ab Initio Molecular Dynamic p.29/67

51 Isotropic NPT : Hoover Extended system, similar to the Nosé-Hoover thermostat r = V 1/3 s si = pi m i V 1/3 ξ = gk B Q χ = V 3V (T T) pi = F i (ξ + χ) p i χ = P(t) P τ 2 P k BT with g number of degrees of freedom, Q mass of the thermostat, τ P relaxation time of the pressure Ab Initio Molecular Dynamic p.30/67

52 Anisotropic NPT Parrinello-Rahman Idea : reduced variable r i = h s i, with h = [ a, b, c ] and a, b et c cell parameters We introduce G = h t h P pressure tensor : ( P αβ = 1 m i r iα r iβ + ) r ijα f ijβ V i j i Hamiltonian H = K Q α β ḣ 2 αβ + V + PV Ab Initio Molecular Dynamic p.31/67

53 Anisotropic NPT Parrinello-Rahman Idea : reduced variable r i = h s i We introduce G = h t h Hamiltonian H = K Q ḣ 2 αβ + V + PV α Equations of motion m s = h 1 f mg 1 Ġ s QḦ = (P 1P)V ( h 1) t β with Q mass of the box Ab Initio Molecular Dynamic p.31/67

54 Equilibration Listening phase We check that variables that should be conserved are stable, no drift : T E RMSD : Root Mean square displacement g(r) : pair distribution function One simple criterion : averages should not depend on the initial time. Ab Initio Molecular Dynamic p.32/67

55 Analysis Production phase Compute the interesting properties : RMSD Structural parameters g(r) Fluctuations C v Vibrational spectra Kinetic constant Average on a given window, larger than fluctuation time (see Practical session!) Thermostat, Barostat are perturbing the averages Ab Initio Molecular Dynamic p.33/67

56 Shall we go beyond classical MD Pros of classical MD : Fast Good geometries, vibrations, transport properties Thermodynamical properties accurate for parameterized systems Cons Parameters! One need to know how atoms are linked : Chemical reactions cannot be studied by standard FF ReaxFF, ReBO, LOTF... No excited states No electronic properties : charge, density,... Ab Initio Molecular Dynamic p.34/67

57 Ab Initio Molecular Dynamics Ab Initio Molecular Dynamic p.35/67

58 Which ab initio MD? Electron explicitly taken into account Let us denote by : M k, Z k, R k the mass, the charge and the position of a nucleus R k ou v k the velocity of a nucleus pk = M k vk the impulsion of a nucleus m e, r i the mass and the position of an electron Ab Initio Molecular Dynamic p.36/67

59 Which ab initio MD? Electron explicitly taken into account We would like to solve φ i with ({ Rk }, { ) r i } ;t t = H ({ Rk }, { ) ({ Rk } r i } φ, { ) r i } ;t H = electrons i i 2 } {{ } T e élec. nuclei Z k elec. elec. 1 Z k Z l R kl + + r i ik r k i j>i ij k l>k }{{} V ne noy. k 2M k k }{{} Ab Initio Molecular Dynamic T N p.36/67

60 TDSCF approach Hypothesis : ({ Rk } φ, { ) r i } ;t Ψ ({ ({ Rk } r i } ;t) χ with the convenient phase factor Ẽ e (t ) = d r i d R k i Ψ ({ r i } ;t)χ ({ Rk } Let us write d r = i d r i and d R = k d R k k ) [ i ;t exp ] dt Ẽ e (t ) t 0 t ) ;t H e Ψ ({ ({ Rk } ) r i } ;t) χ ;t Ab Initio Molecular Dynamic p.37/67

61 TDSCF approach Hypothesis : ({ Rk } φ, { ) r i } ;t Ψ ({ r i } ;t) χ ({ Rk } ) [ i ;t exp ] dt Ẽ e (t ) t 0 t with Ẽ e (t ) = d r d R Ψ ({ r i } ;t)χ ({ Rk } ) ;t H e Ψ ({ ({ Rk } ) r i } ;t) χ ;t Ab Initio Molecular Dynamic p.37/67

62 TDSCF approach Hypothesis : ({ Rk } φ, { ) r i } ;t Ψ ({ r i } ;t) χ ({ Rk } ) [ i ;t exp ] dt Ẽ e (t ) t 0 t we get (project on < Ψ, < χ, use d < H > /dt = 0) : i Ψ t = i i χ t = k iψ + { 1 2M k 2 kχ + { d Rχ ({ Rk } ) ({ Rk } ;t V ne χ ;t) } Ψ d r Ψ ({ r i } ;t) H e Ψ ({ } r i } ;t) χ Ab Initio Molecular Dynamic p.37/67

63 TDSCF approach Let us write (A,S : real) ({ Rk } ) χ ;t ({ Rk } ) = A ;t exp [ is ({ Rk } ) ] ;t / we get, after separating real and imaginary part : i Ψ t = i S t = k A t = k iψ + { 1 2M k ( k S) 2 d Rχ ({ Rk } 1 M k ( k S) ( k A) k d r Ψ H e Ψ + k ) ({ Rk } ;t V ne χ ;t) } Ψ 1 A ( 2 2M ks ) k 1 2 k A 2M k A Ab Initio Molecular Dynamic p.37/67

64 Ehrenfest approach Classical nuclei : = 0 S t + k 1 2M k ( k S) 2 + d r Ψ H e Ψ = 0 Hamilton-Jacobi equation analogy : S t + k 1 2M k (p k ) 2 + V e = 0 Using p k k S M k Rk (t) = k d r Ψ H e Ψ Ab Initio Molecular Dynamic p.38/67

65 Ehrenfest approach Classical nuclei : ({ Rk } ) χ ; t 2 = k d ({ Rχ Rk } Rk ({ Rk } ; t) χ lim 0 ( Rk δ ) R k (t) ) ; t = R k (t) That is i Ψ t = i 2 iψ + V ne Ψ = H e Ψ Ab Initio Molecular Dynamic p.39/67

66 Ehrenfest summary Wavefunction explicitly propagated, coupled to the nuclei M k Rk (t) = k d r Ψ H e Ψ i Ψ t = H eψ No electronic minimization, except for t = 0 Transitions between electronic states are explicitly described t imposed by electrons dynamics Very very small Ehrenfest seldom used Ab Initio Molecular Dynamic p.40/67

67 Born-Oppenheimer Wavefunction is not propagated Time { independent Schrödinger equation is solved for Rk } each : M k Rk (t) = k min Ψ 0 Ψ 0 H e Ψ 0 E 0 Ψ 0 = H e Ψ 0 Electronic minimization at each step No more transitions between electronic states t imposed by the nuclei relatively large Ab Initio Molecular Dynamic p.41/67

68 Born-Oppenheimer : HF, DFT Simplifications using independent electron methods : HF ou DFT-KS Molecular Orbitals denoted by {ϕ i } Energy minimization with orthogonal MO : : L e = Ψ 0 H e Ψ 0 + i,j Λ ij ( ϕ i ϕ j δ ij ) leads to : H eff ϕ i = j Λ ij ϕ j Ab Initio Molecular Dynamic p.42/67

69 Born-Oppenheimer : HF, DFT Simplifications using independent electron methods : HF ou DFT-KS Molecular Orbitals denoted by {ϕ i } Energy minimization with orthogonal MO : leads to : H eff ϕ i = j Λ ij ϕ j Diagonal form : H eff ϕ i = ǫ i ϕ i Ab Initio Molecular Dynamic p.42/67

70 Born-Oppenheimer : HF, DFT Simplifications using independent electron methods : HF ou DFT-KS Molecular Orbitals denoted by {ϕ i } Energy minimization with orthogonal MO : leads to : H eff ϕ i = j Λ ij ϕ j So that : M k Rk (t) = k min Ψ 0 Ψ 0 H e Ψ 0 0 = H eff ϕ i + j Λ ij ϕ j Ab Initio Molecular Dynamic p.42/67

71 Car-Parrinello (1/2) Goal : benefit from all advantages Ehrenfest : No electronic minimization BO : Large time step Tool : Adiabatic separation between fast electrons and slow nuclei How : MO described as classical variables Ab Initio Molecular Dynamic p.43/67

72 Car-Parrinello (1/2) Goal : benefit from all advantages Tool : Adiabatic separation between fast electrons and slow nuclei How : MO described as classical variables, but decoupled from the nuclei Using an extended Lagrangian formulation Ab Initio Molecular Dynamic p.43/67

73 Car-Parrinello (2/2) Goal : benefit from all advantages Tool : Adiabatic separation between fast electrons and slow nuclei How : MO described as classical variables, but decoupled from the nuclei L CP = 1 2 M kr 2 k µ i ϕ i ϕ i k i }{{} Kinetic energy Ψ 0 H e Ψ 0 }{{} Potential energy + constraints }{{} orthogonality, geometric... µ "Fictitious" mass of the electrons Ab Initio Molecular Dynamic p.44/67

74 Car-Parrinello (2/2) Goal : benefit from all advantages Tool : Adiabatic separation between fast electrons and slow nuclei Constant of motion H CP = k 1 2 M kr 2 k + i 1 { 2 µ Rk }) i ϕ i ϕ i + E el ({ϕ i }, Ab Initio Molecular Dynamic p.44/67

75 Car-Parrinello (2/2) Goal : benefit from all advantages Tool : Adiabatic separation between fast electrons and slow nuclei Equations of motion M k Rk (t) = k Ψ 0 H e Ψ 0 + k {constraints} µ ϕ i (t) = δ δϕ i Ψ 0 H e Ψ 0 + δ δϕ i {constraints} t approximately 5 to 10 times smaller than BO Ab Initio Molecular Dynamic p.44/67

76 Car-Parrinello : HF,DFT-KS We use HF or DFT-KS methods L CP = k 1 2 M kr 2 k + i 1 2 µ i ϕ i ϕ i Ψ 0 H eff Ψ 0 + i,j Λ ij ( ϕ i ϕ j δ ij ) Ab Initio Molecular Dynamic p.45/67

77 Car-Parrinello : HF,DFT-KS We use HF or DFT-KS methods Equations of motion become M k Rk (t) = k Ψ0 H eff Ψ 0 µ ϕ i (t) = H eff ϕ i + j Λ ij ϕ j Very similar to BO : µ ϕ i (t) = 0 Ab Initio Molecular Dynamic p.45/67

78 Why does it work? Fast electrons, slow nuclei Electronic frequencies for Si 8 : f e (ω) = Ψ(t) Ψ(0) dt t=0 cos(ωt) i Triangle : latest nuclear frequency Ab Initio Molecular Dynamic p.46/67

79 Why does it work? Electronic Oscillations close to the BO surface E cons = 1 2 M kr 2 k µ i ϕ i ϕ i + Ψ H e Ψ k i E phys = 1 2 M kr 2 k + Ψ H e Ψ = E cons T e k Model System : Si FCC, 2 atoms/cell Ab Initio Molecular Dynamic p.47/67

80 Why does it work? Electronic Oscillations close to the BO surface Model System : Si FCC, 2 atoms/cell Small oscillations, stable in time Ab Initio Molecular Dynamic p.47/67

81 Why does it work? Electronic Oscillations close to the BO surface Model System : Si FCC, 2 atoms/cell Small oscillations, stable in time Ab Initio Molecular Dynamic p.47/67

82 Why does it work? Forces oscillations very small Model System : Si FCC, 2 atoms/cell Small oscillations, stable in time Oscillations averaged to 0 Ab Initio Molecular Dynamic p.48/67

83 Why does it work? Forces oscillations very small Model System : Si FCC, 2 atoms/cell Small oscillations, stable in time Oscillations averaged to 0 Ab Initio Molecular Dynamic p.48/67

84 Working Condition CP : adiabaticity Fast electrons, small nuclei : µ M k ϕ i 0 : wavefunction close to BO, always slightly above. Compromise for µ : t large, but we want electron/nuclei separation Electronic frequency ϕ i occ. and ϕ a virtual : ω e ia = 2(ǫ a ǫ i )/µ Smallest : ω e min E GAP /µ Highest : ω e max E cut /µ so that t max µ/e cut Usually : µ = , t = 5-10 au = fs. Ab Initio Molecular Dynamic p.49/67

85 Loosing adiabaticity If E GAP 0, then ω e min too close to ω nucl For example for an elongated bond, ex Sn 2 Ab Initio Molecular Dynamic p.50/67

86 Loosing adiabaticity If E GAP 0, then ω e min too close to ω nucl For example for an elongated bond, ex Sn 2 For metallic systems! Two solutions : Back to BO Electronic Thermostat Ab Initio Molecular Dynamic p.50/67

87 Comparison BO/CP BOMD CPMD Always on BO surface Always slightly above t τ nucl t τ nucl t BO 5 10 t CP Minimization at each step Only Orthogonalization Problem when deviating Stable with respect to deviations from the BO surface Works for E GAP = 0 Electronic thermostat Used for solid systems Used for liquids Ab Initio Molecular Dynamic p.51/67

88 Comparison BO/CP Ab Initio Molecular Dynamic p.52/67

89 Comparison Ehrenfest/CP Ehrenfest MD CPMD Real separation (quantum) Fictitious separation (classical) t τ elec t τ elec t CP 5 10 t Ehrenfest Rigorous Orthonormality Imposed by constraints Deviations from BO add up Stable Ab Initio Molecular Dynamic p.53/67

90 Conclusion and perspectives AIMD : includes electronic effects, allows for chemical reactions, catalytic processes... CPMD : Very used, very fashion! but... Number of atoms still limited DFT not always sufficient Perspectives Hybrid methods : QM/MM, QM/QM Post-HF and TD approachs : excited states, highly correlated materials... BO? New algorithms faster, more efficient, becomes competitive with CPMD. Ab Initio Molecular Dynamic p.54/67

91 Application : Free energy calculations Ab Initio Molecular Dynamic p.55/67

92 Free energy profile Reaction ({ coordinate q = Rk }) q Probability density P(z) = 1 Q NV T Free energy profile d Rd P exp( βh)δ F(z) = k B T ln P(z) ( q ({ Rk }) ) z Reaction rate : k = k BT e rg RT h Ab Initio Molecular Dynamic p.56/67

93 Rare event? System position for E a = 2 k B T Diffusive system. Ab Initio Molecular Dynamic p.57/67

94 Rare event? System position for E a = 5 k B T Nothing at the TS! Ab Initio Molecular Dynamic p.57/67

95 Rare event? System position for E a k B T Alternating between the two states. Long residence time, short transition time. k TST 1s 1 Slow reaction rare event. Ab Initio Molecular Dynamic p.57/67

96 Why is it so difficult? Standard MD is a "real time" method, t fs Chemical reaction time : fast around ns, typical around µs-ms, biology can be seconds Two incompatible time scales More, phase space dimension is 6N, impossible to fully sample Ab Initio Molecular Dynamic p.58/67

97 Why is it so difficult? Two incompatible time scales Solutions Work at higher T : faster sampling but might be not the same phenomena Force the reaction to occur : Biais potential Constrained dynamic Adiabatic dynamic Metadynamic In all cases, we calculate F, not F Ab Initio Molecular Dynamic p.58/67

98 Bias potential Many approaches : Umbrella sampling, adaptive biais potential, accelerated dynamics, flooding potential... Main Idea : we add a potential to "erase" the barrier V ({ Rk }) = V ({ Rk }) + V Then get the non-biaised energy : O ({ Rk }) = O ({ Rk }) exp (β V) ( q ({ Rk })) exp (β V) Problem : we do not know the barrier position, height and shape! In practice : many simulations with a moving model Ab Initio Molecular Dynamic p.59/67 potential

99 Umbrella Sampling Usual bias : V ( q ({ Rk })) [ q ({ Rk }) ] 2 z Simulations for different z Ab Initio Molecular Dynamic p.60/67

100 Constrained dynamics Two families : q evolves continuously from z 0 to z f during the simulation Slow growth method : slow evolution, quasi-static Fast growth method, Jarzinsky : fast evolution, average taken on many trajectories exp ( βw) = exp ( βf) Many simulation with q set to different values of z. Ab Initio Molecular Dynamic p.61/67

101 Thermodynamic integration Change in free energy calculated by integrating free energy derivative : F(z = z 0 z = z f ) = zf z 0 F q dq Numerical evaluation using discrete values A constrained simulation is launched for each value of q(z). How to obtain F q (z)? Ab Initio Molecular Dynamic p.62/67

102 Free energy derivatives Two problems : q = z q = 0 : no sampling for the impulsion p q. Blue-Moon : O(z) = Z 1/2 O cont Z 1/2 cont with Z the reduce mass associated to q : Z = k q is non linear with respect to J non unity 1 M k q R k q R k { Rk }, Jacobian matrix Ab Initio Molecular Dynamic p.63/67

103 Free energy derivatives Two problems : q = z q = 0 : no sampling { for the impulsion p q. Rk } q is non linear with respect to, Jacobian matrix J non unity F q (z) = V q + k BT ln J q q=z Ab Initio Molecular Dynamic p.63/67

104 Free energy derivatives Two problems : q = z q = 0 : no sampling { for the impulsion p q. Rk } q is non linear with respect to, Jacobian matrix J non unity F q (z) = Z 1/2 [ λ + k B T G] q=z Z 1/2 q=z q R k 2 q q R k R l R l, with G = 1 1 Z k,l 2 M k M l λ Lagrange multiplier associated to the constraint σ ({ Rk }) = q z = 0 (SHAKE) Ab Initio Molecular Dynamic p.63/67

105 Adiabatic dynamics Idea : Decoupling motions along q from other motions Use different temperatures One (chain) thermostat associated to q One (chain) thermostat associated to the N-1 other degrees of freedom T q T nuc Change the mass of atoms concerned by q Ab Initio Molecular Dynamic p.64/67

106 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV L = L 0 + α 1 2 M α s α 2 α 1 2 k α [S α (r) s α ] 2 M α large enough natural adiabatic separation Ab Initio Molecular Dynamic p.65/67

107 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV on top of this, potential wells are filled with gaussians V (t) = i W exp ( s s i(t) 2 2δσ 2 Important choices for W, σ, time interval between adding two gaussians ) Ab Initio Molecular Dynamic p.65/67

108 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV Ab Initio Molecular Dynamic p.65/67

109 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV Ab Initio Molecular Dynamic p.65/67

110 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV Ab Initio Molecular Dynamic p.65/67

111 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV Ab Initio Molecular Dynamic p.65/67

112 Coarse MD Metadynamic Idea : Similar to the adiabatic dynamics, using extended Lagrangian! Reaction coordinates are included as collective variables (CV) s α for CV Ab Initio Molecular Dynamic p.65/67

113 How to choose a good RC? In practice, difficult to sample a coordinate space larger than 3D Chemist intuition might fail!! Ab Initio Molecular Dynamic p.66/67

114 Conclusion Already many tools but choosing RC is still a hot topic for large systems Classical nuclei : what about quantum effects? Corrections a posteriori : tunneling, isotopic effects Path Integral ZPE correction Ab Initio Molecular Dynamic p.67/67