Newtonian and extended Lagrangian dynamics
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1 Newtonian and extended Lagrangian dynamics Gianni Cardini and Riccardo Chelli Dipartimento di Chimica Ugo Schiff Università di Firenze, Via della Lastruccia 3, Sesto Fno, Firenze SMART January 25-29, 2016
2 The Molecular Dynamics Method Numerical experiments on model systems You need: to define a model equations of motion numerical integration A COMPUTER G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58 12
3 It is useful! Molecular dynamics methods provide essentially exact results for a model Test of theories in controlled conditions Test of a model MD can suggest new experiments Microscopic description G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58 13
4 How to? Model Ṣtucture, conditions, potential Rule to solve the equations of motions ṇumerical integration alghorithm, time step analysis {r i (t)} {v i (t)} G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58 14
5 Reminder Molecular dynamics gives exact results on a model If the model is wrong the results are wrong! If you ask a wrong question you will receive a wrong answer! G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58 15
6 Microscopic description of a chemical system Initial position and momentum of the particles Interaction law No nuclear quantum effects Classical mechanics No electronic transition Ḅorn Oppenheimer approximation G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58 16
7 Interaction law, Analitical potential V = v 1 (r i ) + v 2 (r i, r j ) + i ij ijk v 4 (r i, r j, r k, r l ) + ijkl one body:v 1 (r i ) ẹxternal field two bodies:v 2 (r i, r j ) ạtom-atom, electrostatics, stretching three bodies:v 3 (r i, r j, r k ), bending four bodies:v 4 (r i, r j, r k, r l ), torsion v 3 (r i, r j, r k )+ 17 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
8 NE ensemble Isolated system Clusters Newtonian mechanics Ẹquation of motion from II law: F = ma = r V (r) Constraints Ḥolonomic constraints [f ({r}, t) = 0] Lagrangian mechanics Ḷ({q}, { q}) = K V two scalars! G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58 18
9 Lagrangian N particles L({q}, { q}) = K V = 1 2 N j=1,n Equations of motion d L L = 0 dt q j q j M holonomic constraints: f k ({r}, t) = 0 M G j = λ k j f k ({r}, t) λ k Lagrange multipliers k=1 m i q 2 j V ({q}) d L L = G j dt q j q j 19 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
10 Hamiltonian From Lagrangian p = L(q, q) q Equations of motion H(q, p) = dq dt p = K + V dq dt dp dt = H p = H q 6N first order diff eq instead of 3N second order G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
11 Condensed phase simulations NVE ensemble Surface effects Periodic Boundary Conditions Angular Momentum no more conserved Spurious correlations ṃinimum image PBC & Electrostatics:Ewald V (ϵ = 1) = V d + V r + V s + V shape V d = 1 2 N i=1 j=1 N erfc(α rij + R ) q i q j r ij + R R=0 V r = 1 2 N i=1 j=1 V s = α N π i=1 q2 i N q i q j (4π/Λ) k 0 1 k 2 exp( k 2 /4α 2 ) cos(k r ij ) V shape = 2π 3Λ N i=1 q 2 ir i G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
12 Other Ensembles P=costant HC, J Chem Phys 72 (1980) 2384 (NPH) M Parrinello, A Rahaman, Phys Rev Lett 45 (1980) 1196 T=costant HC, J Chem Phys 72 (1980) 2384 (MC/MD) S Nosé, Mol Phys (1984) 255 () Extended Lagrangians New dynamical variables are added L = L + K (new v) V (new v) G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
13 Numerical Integration Choice of the time step, t Large as possible ṭo sample the phase space with less steps as possible but sufficiently small to sample the fastest motion and to obtain an acceptable conservation of the constants of the motion G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
14 Numerical integrators and Lioville operators Time evolution of f (x) x ({q, p}) phase space df 3N [ f dt = H f ] H q n p n p n q n n=1 Liouville operator: L 3N ıl = n=1 [ H p n H q n q n p n ] ılf (x) = df (x) dt G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
15 Numerical integrators and Lioville operators formal solution f (x t ) = e ılt f (x 0 ) classical propagator: e ılt ; qm propagator e ıht/ħ Starting point for approximate solutions ıl = ıl 1 + ıl 2 do not commute ıl 1 = ıl 2 = 3N H p n n=1 3N n=1 q n H q n [ıl 1, ıl 2 ] 0 p n 115 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
16 Numerical integrators and Lioville operators e ılt e ıl 1t e ıl 2t ọften the action of e ıl 1t and e ıl 2t can be evaluated exactly symmetric Trotter theorem (1959) given two operators such that [A, B] 0 then ) e (A+B)t = lim (e B t 2m e A t m e B t m 2m m Applying the symmetric Trotter theorem e ılt = lim (e ıl 2 t t 2m e ıl 1 m ) t m m e ıl 2 2m G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
17 Numerical integrators and Lioville operators posing t = t m e ılt = ) lim (e ıl 2 t 2 e ıl 1 t e ıl 2 t m 2 m, t 0 Hans de Raedt e Bart de Raedt: Approximate propagation [ Phys Rev A28 ( 1983) ] Tuckerman et al ṛeversible integrators e ıl t e ıl 2 t 2 e ıl 1 t e ıl 2 t 2 + O( t 3 ) G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
18 Numerical integrators and Liouville operators Example: one dimensional Hamiltonian H = p2 2m + V (q) ıl 2 = F (q) p eıl 2 t/2 = e t 2 F (q) p ıl 1 = p m q eıl1 t = e t p m q Exponential of operators e a t = (a t) k k=0 k! G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
19 Numerical integrators and Liouville operators First operator e t ( ) 2 F(q) p q(0) = p(0) k=0 ( = momentun translation Second operator e t p m ( ) q q(0) p( t 2 ) = position translation ( 1 t k! 2 F (q) ) k ( ) q(0) p p(0) ) ( ) q(0) = p ( ) t 2 q(0) p(0) + t 2 F (q) ( q(0) + t p ) m p( t 2 ) = ( ) q( t) p( t 2 ) G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
20 Numerical integrators and Lioville operators Position is changed then recompute the force Third operator ( ) ( ) e t 2 F (q( t)) p q( t) q( t) p( t 2 ) = p( t 2 ) + t 2 ( ) F (q( t)) q( t) = p( t) momentum translation This is Warning When the system is subject to olonomic constraints (SHAKE, RATTLE ) the Lagrange multipliers have to be determined to full convergence G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
21 Complex systems different time scale Potential Energy Atomic Forces U({r i }) = U intra + U inter F i = r i U = i U intra i U inter = F intra i Internal forces fast motions Intermolecular forces slow motions + F inter i G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
22 System characterized by fast and slow motions Liouville operator ıl = ıl fast + ıl slow ıl fast = p m ıl slow = F slow p Reference Hamiltonian q + Ffast p H ref = p2 2m + U(q)fast F fast (q) = du(q)fast dq q = p m fast G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
23 RESPA, reference system propagator alghorithm Tuckerman et al JCP 97(1992) 1990 Factorization of the full propagator Defining τ = t n e ıl t t ılslow = e 2 e ıl fast t t ılslow e 2 e ılfast t = [ ] e τ 2 Ffast (q) p e τ p m q e τ 2 Ffast (q) n p G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
24 full RESPA propagator e ıl t =e t 2 F slow (q) p [ ] e τ 2 F fast (q) p e τ p m q e τ 2 F fast (q) n p e t 2 F slow (q) p G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
25 Pseudocode p = p + 05 * dt * Fslow do j=1,n p = p + 05 * dt/n * Ffast q = q + dt/n * p/m! modified coordinates call FastForce p = p + 05 * dt/n * Ffast enddo! coordinates call SlowForce p = p + 05 * dt * Fslow! momenta G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
26 Canonical Ensemble N V T System in contact with an infinite thermal bath The H of the system is not conserved The H follows a Boltzmann distribution e βh G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
27 Canonical Ensemble Nosé Lagrangian SNosé, MolPhys 52 (1984) 255 A new degree of freedom: s s describes the interaction with an external bath by a velocity scaling Extended Lagrangian N m L Nose = i s 2 ( ) 2 dri U({r}) 2 dt i + Q ( ) ds 2 (3N + 1)kT eq ln(s) 2 dt G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
28 Canonical Ensemble Nosé Hamiltonian conjugate momenta from the Lagrangian p i L ṙ i = m i s 2 ṙ i p s L ṡ = Qṡ Hamiltonian of the extended system (N particles + s) H Nose = N i p 2 i 2m i s 2 U({r}) + p2 s 2Q (3N + 1)kT eq ln(s) G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
29 Canonical Ensemble Nosé equations of motion virtual variables:p, r, t dr i dt dp i dt ds dt dp s dt = H dp i = = H dr i p i m i s 2 = U({r}) r i = H = p s dp s Q = H ds = ( N i p 2 i m i s 2 (3N + 1)kT eq ) 1 s 129 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
30 Canonical Ensemble real variables r and s correspond to the real variables, while π = p/s, p s = p s /s and τ = t/s equations of motion dr i dτ = s dr i dt d π i dτ = s dp i/s dt ds dτ = s ds dt dsp s/q dτ = s Q dp s dt = p i m i s = π i m i = dp i dt = sp s Q ( N = i 1 s p i ds dt π 2 i m i (3N + 1)kT eq ) 1 Q G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
31 Canonical Ensemble Constant of the motion in real variables H Nose = N i + (sp s) 2 2Q π 2 i 2m i + U({r}) + (3N + 1)kT eq ln s transformation to real variables is not canonical H Nose It is not an Hamiltonian implementation in real variables eqm are not easy to be implemented in MD G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
32 Canonical Ensemble equations Hoover PhysRevA31 (1985)1695: non canonical change of variables in Nosé eqm π i = p i dt s,dτ = s, 1 ds s dτ = dη dt and p s = p η and posing the number of dof to 3N ṙ i = p i m ṗ i = i U({r}) p η Q p i η = p η Q N p 2 i ṗ η = 3NkT eq m i i G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
33 Canonical Ensemble η acts as a friction term ṗ η twice the difference of kinetic energy and its canonical average It is a non Hamiltonian system Conserved Energy H(r, p, η, p η ) = H(r, p) + p2 η 2Q + 3NkT eq The real Hamiltonian is H(r, p) When N i F i = 0 also e η N i F i is constant and the distribution function of the momenta is wrong f (p) 1 2πmkT e p2 2mkT 133 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
34 Canonical Ensemble Martyna et al J Chem Phys 97 (1992) 2635 The wrong distribution function is due to the existence of two conservation laws The eqm do not have a sufficient number of variables! Chain The f (p η ) must follow a Maxwell-Boltzmann distribution p η must be coupled to a thermostat The chain of thermostats should be infinite The lenght is chosen finite (M thermostats) G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
35 Canonical Ensemble Chain eqm ṙ i = p i m ṗ i = F i p η 1 Q 1 p i η j = p η j ṗ ηm = Q j ṗ η1 = N i=1,n j = 1,, M p 2 i m i 3NkT [ p 2 ηj 1 ṗ ηj = kt Q j 1 [ p 2 ηm 1 kt Q M 1 ] p η 2 Q 2 p η1 p η j+1 p ηj j = 2,, M 1 Q j+1 ] 135 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
36 Canonical Ensemble Chain Previous equations cannot be transformed in an Hamiltonian system Optimal choice: Q 1 = 3NkT τ 2, Q j = NkT τ 2 j = 2,, M τ > 10 t Conserved Energy H = H(r, p) + M j=1 p 2 η j 2Q j + 3NkT η 1 + kt M j=2 η j G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
37 Canonical Ensemble Massive Chain add a separate chain to each atom of the system much more rapid equilibration some modes are often weakly coupled Tobias et al J Phys Chem 97 (1993) rapid thermalization of a protein in solution G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
38 Canonical Ensemble Integration of Chain eqm The Liouville operator can be written as: ıl = ıl NHC + ıl 1 + ıl 2 e ıl t = e ılnhc t/2 e ıl2 t/2 e ıl1 t e ıl2 t/2 e ılnhc t/2 +O( t 3 ) e ıl H t = e ıl 2 t/2 e ıl 1 t e ıl 2 t/2 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
39 Canonical Ensemble Thermostat forces G 1 = N i=1,n p 2 i G j = p2 η j 1 Q j 1 kt 3NkT m i ıl NHC = + M 1 j=1 N i=1 p η1 p i + Q 1 p i ( G j p ηj p ηj+1 Q j+1 M j=1 p ηj Q j η j ) p ηj + G M p ηm G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
40 Canonical Ensemble Factorization of the propagator is not sufficient! The resulting integrator is not sufficiently robust The application of RESPA alone require many cycles An improvement has been obtained coupling RESPA with an higher order fatorization scheme: Suzuki-Yoshida [Phys Lett A150(1990)262,J Math Phys 32(1991)400] G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
41 Canonical Ensemble Suzuki-Yoshida Given a primitive factorization, S(λ), of an operator S(λ) = e λa 2/2 e λa 2 e λa 1/2 = e λ(a 1+A 2 ) chosen an even order of the error in the factorization 2s this gives n SY = 5 s 1 weigths, w α, such that: n SY w α = 1 α=1 The factorization is e λ(a 1+A 2 ) n SY α=1 S(w α λ) G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
42 Canonical Ensemble RESPA Setting λ = t/2 e ıl NHC t/2 n SY α=1 S(w α t/2) the operator is to be applied n times with a time step w α t 2n e ıl NHC t/2 [ nsy α=1 ( ) ] n t S w α 2n G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
43 Canonical Ensemble Factorization of e ıl NHC t/2 The choice is not unique A new kind of operator e cx x Given a function p, the action of the operator is [ e cp c k ( p p = p ) ] k p k! p k=0 = p k=0 c k k! = pec Given a function f (p), the results is f (pe c ) Terms of this kind will act scaling the momentum of the thermal bath G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
44 Isoenthalpic-Isobaric Ensemble HC, JChem Phys 72 (1980) 2384 First attempt to perform MD simulations at costant P The volume,v, is treated as a dynamical variable Cartesian coordinates {r i } are replaced by scaled coordinates { ρ i }: ρ i = r i V 1/3 Postulated Lagrangian: L({ ρ}, { ρ}, V, V ) = 1 N 2/3 mv 2 ρ i ρ i i=1 N i<j=1 U(V 1/3 ρ ij ) M P V 2 P ext V with M P piston mass and P ext external pressure G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
45 Isoenthalpic-Isobaric Ensemble Hamiltonian H = 1 2mV 2/3 N i= M P Π 2 + P ext V real variables Conserved Energy N 1 π i 2 + i=1 N U(V 1/3 ρ ij ) j>i with p V = M P V H = H 0 (r, p) + p2 V 2M P + P ext V G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
46 Isoenthalpic-Isobaric Ensemble eqm Martyna, Tobias, Klein JChemPhys 101(1994)4177 a variable ϵ = 1 3 ln(v /V V 0) with momentum p ϵ = M P 3V ṙ i = p i m i + p ϵ M P r i ṗ i = F i p ϵ M P p i V = 3Vp ϵ M P ( 1 + ṗ ϵ = 3V (P int P ext ) + 3 N dof ) 3 (3N N constr ) N p 2 i m i i=1 with P int = 1 3 Tr[Pint αβ ] and Pint αβ = 1 [ ] N piα p iβ V i=1 m i + F iα r iβ 146 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
47 Isoenthalpic-Isobaric Ensemble Parrinello - Rahman JApplPhys 52(1981)7182 Parrinello and Rahaman same estended Lagrangian of but with variable cell shape transformation matrix from scaled,s, to cartesian,r a x b x c x H = a y b y c y a z b z c z r = Hs real distances by metric matrix G = H t H V = det H, cell sides a = ax 2 + ay 2 + az 2 9 elements of the matrix are used as dynamical variables 3 sides and 3 angles of the cell 3 Euler angles!! 147 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, / 58
48 Isoenthalpic-Isobaric Ensemble Molecular case Nosé and Klein Mol Phys 50 (1983) 1055 No rotating simulation box H 11 H 12 H 13 H = 0 H 22 H H 33 H 11 = a H 12 = b cos(γ) H 13 = c cos(β) H 22 = b sin(γ) H 23 = cos(α) cos(β) cos(γ) c sin(γ) H 33 = c 2 H13 2 H2 23 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
49 Isoenthalpic-Isobaric Ensemble Molecular case Center of mass scaling Small molecules center of mass variables: R α ; P α ; F α P internal molec = 1 3V Molec α=1 [ ] P 2 α + R α M F α α G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
50 Isobaric Ensemble Phys Rev B44 (1991) 2358 Dynamical variable Stress Tensor Very useful to optimize crystal structure G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
51 es Molecules { N n } L = s 2 m iα 2 σ iα 2 + M α 2 s2 (H ρ α ) t (H ρ α ) α=1 i=1 U({ σ iα + H ρ α }) + W P 2 Tr(Ḣt Ḣ) + W s 2 ṡ2 P ext det(h) 3NkT ext ln s Lagrangian with a simple Nosé Nosé-Hoover chain separated thermostats for atoms and barostat G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
52 factorization Tuckerman et al J Phys A39 (2006) 5629 factorization scheme for an atomic system e ıl t = e ıl NHC baro t/2 e ıl NHC part t/2 e ıl g,2 t/2 p g = W p ḢH 1 ıl 1 = ıl 2 = e ıl 2 t/2 e ıl g,1 t e ıl 1 t e ıl 2 t/2 e ıl g,2 t/2 e ıl NHC part t/2 e ıl NHC baro t/2 N i=1 N i=1 [ pi + p ] g r i m i W p [ ( F i p g W p + 1 ) ] Tr[p g ] I p i N f W P p i G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
53 ıl g,1 = p gh W p H [ ıl g,1 = ıl NHC part = M 1 j=1 det[h](p int IP ext ) + 1 N f N i=1 p η1 p i + Q 1 p i ( G j p ηj p ηj+1 Q j+1 M j=1 p ηj Q j N i=1 η j p 2 i m i I ) p ηj + G M p ηm ] p g ıl NHC baro = same as ıl NHC part with p g G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
54 pseudo masses For T of particles: For T of barostat For pressure Q 1 = 3kT τ 2 p Q 1 = kt τ 2 b Q j = kt τ 2 p Q j = kt τ 2 b Martyna 1992, 1996 W P = (3N + 3)kT τ 2 b G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
55 First Principles Molecular Dynamics: Unified Approach for Molecular Dynamics and Density-Functional Theory R Car and M Parrinello Phys Rev Lett 55 (1985) 2471 extended Lagrangian L = + ij occorb i µ i ψ i (r) 2 dr + 1 M α Ṙ 2 α E[{ψ i }, R α ] 2 α ( ) λ ij ψi (r)ψ i(r)dr δ ij An intertial factor µ i, pseudomass (mass lenght 2 ), is associated to the electronic degree of freedom G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
56 First Principles Molecular Dynamics: : equations of motion Assumption: the system is on the BO surface µ ψ i = δe occorb + λ ij ψ j i = 1,, occorb (1) δ ψ i Hellman-Feynman j=1 M I RI = RI E I = 1,, Atoms (2) RI E({R I }) = Ψ 0 RI H Ψ 0 G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
57 First Principles Molecular Dynamics: : trajectories The trajectories generated by CP equations does not corresponds to the true ones unless E[{ψ i }, {R I }] is in the minimum respect {ψ i } at each time step This is obtained by choosing the value of µ to obtain a decoupling between electronic and nuclear degree of freedom G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
58 Rare events Metadynamics Laio, Parrinello PNAS 99(2002) Iannuzzi, Laio, Parrinello L 90(2003) Estended Lagrangian collective variables s α L MTD = L + α 1 2 M αṡ 2 α α 1 2 k α[s α s α ] 2 V (t, s) k α is the force constant that couple the collective variables to the system V (t, s) is a time dependent potential arising from the accumulation of repulsive gaussian hills modified every MD steps G Cardini, R Chelli (UFirenze) MD SMART January 25-29, /
7 To solve numerically the equation of motion, we use the velocity Verlet or leap frog algorithm. _ V i n = F i n m i (F.5) For time step, we approxim
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