Newtonian and extended Lagrangian dynamics

Newtonian and extended Lagrangian dynamics

Gianni Cardini and Riccardo Chelli Dipartimento di Chimica “Ugo Schiff” Università di Firenze, Via della Lastruccia 3, 50019 Sesto F.no, Firenze [email protected], [email protected]

SMART January 25-29, 2016

Introduction

Introduction

The model

The Molecular Dynamics Method

The simulation Integrators Velocity Verlet Multiple time step

NVT

. Numerical experiments on model systems . You need:

Nosé - Hoover

MTK Constant Pressure Andersen MTK

• to define a model

PR Nosé Klein

• equations of motion

Wentzcovitch

NPT

• numerical integration

MTK NPT

CP-MD

. • A COMPUTER

1.2

G. Cardini, R. Chelli (U.Firenze)

MD


2 / 58

Introduction

It is useful!

The model The simulation Integrators Velocity Verlet Multiple time step

NVT

.

Molecular dynamics methods provide essentially exact results for a model

.

• Test of theories in controlled conditions • Test of a model

Constant Pressure Andersen MTK PR Nosé Klein

• MD can suggest new experiments

.

Nosé - Hoover

MTK

• Microscopic description

Wentzcovitch

NPT MTK NPT

CP-MD

1.3


MD


3 / 58

Introduction

How to?


. Model . .Stucture, conditions, potential

NVT Nosé - Hoover

MTK

. Rule to solve the equations of motions . .numerical integration alghorithm, time step {ri (t)} ∧ {vi (t)} . analysis .

Constant Pressure Andersen MTK PR Nosé Klein Wentzcovitch

NPT MTK NPT

CP-MD

1.4


MD


4 / 58

Introduction

Reminder


NVT

. 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!

Nosé - Hoover

MTK Constant Pressure Andersen MTK PR Nosé Klein Wentzcovitch

NPT MTK NPT

CP-MD

1.5


MD


5 / 58

Introduction

The model

The model The simulation Integrators Velocity Verlet

. Microscopic description of a chemical system . • Initial position and momentum of the particles .

• Interaction law

Multiple time step

NVT Nosé - Hoover


. No nuclear quantum effects . .Classical mechanics

PR Nosé Klein Wentzcovitch

NPT MTK NPT

. No electronic transition . Born Oppenheimer approximation .

CP-MD

1.6


MD


6 / 58

Introduction

Interaction law, Analitical potential V =

∑

v1 (ri ) +

i

∑

∑

v2 (ri , rj ) +

The model

∑

ij

ijk

The simulation

v3 (ri , rj , rk )+

Integrators Velocity Verlet Multiple time step

NVT

v4 (ri , rj , rk , rl ) + · · ·

Nosé - Hoover

MTK

ijkl

Constant Pressure

. one body:v1 (ri ) . .external field

Andersen MTK PR Nosé Klein Wentzcovitch

. two bodies:v2 (ri , rj ) . atom-atom, electrostatics, stretching . . three bodies:v3 (ri , rj , rk ), bending .

NPT MTK NPT

CP-MD

. four bodies:v4 (ri , rj , rk , rl ), torsion . 1.7


MD


7 / 58

Introduction

The simulation

The model

NE ensemble

The simulation Integrators

. Isolated system . C . lusters

Velocity Verlet Multiple time step

NVT Nosé - Hoover

. Newtonian mechanics . Equation of motion from II law: F = ma = −∇r V (r) .

MTK Constant Pressure Andersen MTK PR Nosé Klein

. Constraints . Holonomic constraints [f ({r}, t) = 0] .

Wentzcovitch

NPT MTK NPT

CP-MD

. Lagrangian mechanics . ˙ =K −V L({q}, {q}) . two scalars! 1.8


MD


8 / 58

Introduction

Lagrangian N particles

The model

N 1 ∑ ˙ =K −V = L({q}, {q}) mi q˙ 2j − V ({q}) 2 j=1,N


NVT

. Equations of motion .

Nosé - Hoover

MTK

d ∂L ∂L − =0 dt ∂ q˙ j ∂qj

. . M holonomic constraints: fk ({r}, t) = 0 . M ∑ Gj = − λk ∇j fk ({r}, t)


NPT MTK NPT

CP-MD

k =1

λk Lagrange multipliers d ∂L ∂L − = Gj dt ∂ q˙ j ∂qj

.

1.9


MD


9 / 58

Introduction

Hamiltonian

The model The simulation

. From Lagrangian .


NVT

˙ ∂L(q, q) p= ∂ q˙ .

−→

dq H(q, p) = ·p=K +V dt

Nosé - Hoover

MTK Constant Pressure Andersen

. Equations of motion .

MTK PR Nosé Klein Wentzcovitch

NPT

dq ∂H = dt ∂p ∂H dp =− dt ∂q

.

MTK NPT

CP-MD

6N first order diff. eq. instead of 3N second order

1.10


MD


10 / 58

Introduction

Condensed phase simulations

The model

NVE ensemble

Surface effects −→ Periodic Boundary Conditions Angular Momentum no more conserved . Spurious correlations . .minimum image

The simulation

. PBC & Electrostatics:Ewald . V (ϵ = 1) = Vd + Vr + Vs + Vshape

Constant Pressure

i=1 j=1

Nosé - Hoover

MTK


NPT MTK NPT

CP-MD

R=0

∞

∑ 1 1 ∑∑ qi qj (4π/Λ) exp(−k 2 /4α2 ) cos(k · rij ) 2 k2 N

Vr =

NVT

MTK

∞ ′

N


Andersen

∑ erfc(α∥rij + R∥) 1 ∑∑ Vd = qi qj 2 ∥rij + R∥ N

Integrators

N

i=1 j=1

Vs = − √απ .

∑N

2 i=1 qi


k̸=0

Vshape =

2π 3Λ MD

∑

2

N

i=1 qi ri

1.11


11 / 58

Introduction

Other Ensembles


. P=costant . HC.Andersen, J. Chem. Phys 72 (1980) 2384 (NPH) M. Parrinello, A. Rahaman, Phys. Rev. Lett. 45 (1980) 1196. .


NVT Nosé - Hoover

MTK Constant Pressure

. T=costant . HC.Andersen, J. Chem. Phys 72 (1980) 2384 (MC/MD) S. . Nosé, Mol. Phys. (1984) 255 (NVT) . Extended Lagrangians . New dynamical variables are added .


NPT MTK NPT

CP-MD

L = L′ + K (new v) − V (new v)

1.12


MD


12 / 58

Introduction

Numerical Integration

The model

Choice of the time step, ∆t


NVT Nosé - Hoover

. Large as possible . to . sample the phase space with less steps as possible

MTK Constant Pressure Andersen MTK PR

. but sufficiently small . to sample the fastest motion and to obtain an acceptable conservation of the constants of the motion .

Nosé Klein Wentzcovitch

NPT MTK NPT

CP-MD

1.13


MD


13 / 58

Numerical integrators and Lioville operators

.

df = dt

n=1


. Time evolution of f (x) . x ≡ ({q, p}) ∈ phase space 3N [ ∑

Introduction


NVT

∂f ∂H ∂f ∂H − ∂qn ∂pn ∂pn ∂qn

]

Nosé - Hoover


. Liouville operator: L . ] 3N [ ∑ ∂H ∂ ∂H ∂ ıL = − ∂pn ∂qn ∂qn ∂pn n=1 . ıLf (x) =


NPT MTK NPT

CP-MD

df (x) dt 1.14


MD


14 / 58

Introduction

Numerical integrators and Lioville operators . formal solution . f (xt ) = eıLt f (x0 ) .


classical propagator: eıLt ; qm propagator e−ıHt/ℏ Starting point for approximate solutions . ıL = ıL1 + ıL2 .

NVT Nosé - Hoover


ıL1 =

3N ∑

Nosé Klein

∂H ∂ ∂pn ∂qn

n=1 3N ∑

ıL2 = − . . do not commute .

n=1

Wentzcovitch

NPT MTK NPT

CP-MD

∂H ∂ ∂qn ∂pn

[ıL1 , ıL2 ] ̸= 0

.

1.15


MD


15 / 58

Introduction



. ıLt e ̸= eıL1 t eıL2 t . often the action of eıL1 t and eıL2 t can be evaluated exactly . . symmetric Trotter theorem (1959) . given two operators such that [A, B] ̸= 0 then ( t )m t t e(A+B)t = lim eB 2m eA m eB 2m m→∞ . . Applying the symmetric Trotter theorem . )m ( t t t eıLt = lim eıL2 2m eıL1 m eıL2 2m m→∞ .


NVT Nosé - Hoover


NPT MTK NPT

CP-MD

1.16


MD


16 / 58

Introduction



.

Integrators

posing ∆t =

.

eıLt = .

Velocity Verlet

t m

Multiple time step

lim

m→∞,∆t→0

( ) ∆t ∆t m eıL2 2 eıL1 ∆t eıL2 2

NVT Nosé - Hoover


. Hans de Raedt e Bart de Raedt: Approximate propagation . [ Phys. Rev. A28 ( 1983) 3575-3580 ]


NPT ∆t

.

eıL∆t ≈ eıL2 2 eıL1 ∆t eıL2

∆t 2

+ O(∆t 3 )

MTK NPT

CP-MD

. Tuckerman et al. . reversible integrators .

1.17


MD


17 / 58

Numerical integrators and Liouville operators . Example: one dimensional Hamiltonian . p2 H= + V (q) 2m .

Introduction The model The simulation Integrators Velocity Verlet Multiple time step

NVT Nosé - Hoover

MTK Constant Pressure ∆t ∂ ∂ F (q) ∂p ıL2 = F (q) −→ eıL2 ∆t/2 = e 2 ∂p p ∂ ∆t p ∂ ıL1 = −→ eıL1 ∆t = e m ∂q m ∂q


NPT MTK NPT

CP-MD

. Exponential of operators . e .

a∆t

=

∞ ∑ (a∆t)k k =0

k! 1.18


MD


18 / 58

Introduction

Numerical integrators and Liouville operators

The model

. First operator . e

∆t ∂ F (q) ∂p 2

The simulation Integrators

( ) ∑ ( ) ( ) ∞ 1 ∆t ∂ k q(0) q(0) = F (q) p(0) p(0) k! 2 ∂p k=0 ( ) ( ) q(0) q(0) ( ) = = p(0) + ∆t p ∆t 2 F (q) 2


NVT Nosé - Hoover


.momentun translation

NPT

. Second operator . ( ) ( ( ) p) p ∂ q(0) q(0) + ∆t m q(∆t) ∆t m ∂q e = = p( ∆t p( ∆t p( ∆t 2 ) 2 ) 2 )

MTK NPT

CP-MD

.position translation 1.19


MD


19 / 58

Introduction


The model

Position is changed then recompute the force . Third operator . ( ) ( ) ∆t ∂ q(∆t) q(∆t) F (q(∆t)) ∂p 2 e = ∆t p( ∆t p( ∆t 2 ) 2 ) + 2 F (q(∆t)) ) ( q(∆t) = p(∆t)


NVT Nosé - Hoover


NPT

.momentum translation

MTK NPT

CP-MD

This is Velocity Verlet Warning When the system is subject to olonomic constraints (SHAKE, RATTLE ...) the Lagrange multipliers have to be determined to full convergence. 1.20


MD


20 / 58

Introduction

Multiple time step


Complex systems −→ different time scale Potential Energy

NVT Nosé - Hoover

MTK

U({ri }) = U intra + U inter

Constant Pressure Andersen MTK PR

Atomic Forces


∂ Fi = − U = −∇i U intra − ∇i U inter = Fintra + Finter i i ∂ri

NPT MTK NPT

CP-MD

Internal forces −→ fast motions Intermolecular forces −→ slow motions

1.21


MD


21 / 58

Multiple time step . System characterized by fast and slow motions . Liouville operator


ıL = ıLfast + ıLslow p ∂ ∂ ıLfast = + Ffast m ∂q ∂p ∂ ıLslow = Fslow ∂p

. . Reference Hamiltonian .

NVT Nosé - Hoover


NPT MTK NPT

CP-MD

p2 + U(q)fast 2m dU(q)fast Ffast (q) = − dq p q˙ = m Href =


fast

MD

1.22


22 / 58

Introduction

Multiple time step


. RESPA, reference system propagator alghorithm . Tuckerman et al. JCP 97(1992) 1990 Factorization of the full propagator

NVT Nosé - Hoover


e

.

ıL∆t

=e

ıLslow ∆t 2

e

ıLfast ∆t ıLslow ∆t 2

PR

e


.

Defining τ =

.

∆t n

fast ∆t

.

eıL

NPT MTK NPT

[ τ fast ∂ ]n p ∂ τ fast ∂ F (q) ∂p τ m F (q) ∂p = e2 e ∂q e 2

CP-MD

1.23


MD


23 / 58

Introduction

Multiple time step


NVT

. full RESPA propagator .

Nosé - Hoover


e

ıL∆t

∆t slow ∂ F (q) ∂p 2

Andersen

=e [ τ fast ∂ ]n p ∂ τ fast ∂ F (q) ∂p τ m F (q) ∂p 2 ∂q 2 e e e


NPT

.

e

∆t slow ∂ F (q) ∂p 2

MTK NPT

CP-MD

1.24


MD


24 / 58

Introduction

Multiple time step

The model The simulation Integrators

. Pseudocode .


NVT

p = p + 0.5 * dt * Fslow do j=1,n p = p + 0.5 * dt/n * Ffast q = q + dt/n * p/m ! modified coordinates call FastForce p = p + 0.5 * dt/n * Ffast enddo ! coordinates call SlowForce p = p + 0.5 * dt * Fslow ! momenta

Nosé - Hoover


NPT MTK NPT

CP-MD

1.25


MD


25 / 58

Introduction

Canonical Ensemble


NVT Nosé - Hoover

. NVT . System in contact with an infinite thermal bath. The H of the system is not conserved. The H follows a Boltzmann distribution e−βH .


NPT MTK NPT

CP-MD

1.26


MD


26 / 58

Introduction

Canonical Ensemble


. Nosé Lagrangian . S.Nosé, Mol.Phys. 52 (1984) 255 A new degree of freedom: s s describes the interaction with an external bath by a velocity scaling. .


NVT Nosé - Hoover


. Extended Lagrangian .

.


NPT

( ) N ∑ mi s2 dri 2 LNose = − U({r}) 2 dt i ( ) Q ds 2 + − (3N + 1)kTeq ln(s) 2 dt

MTK NPT

CP-MD

1.27


MD


27 / 58

Introduction

Canonical Ensemble


. Nosé Hamiltonian . conjugate momenta from the Lagrangian


NVT Nosé - Hoover

MTK

∂L = mi s2 r˙ i ∂ r˙ i ∂L ps ≡ = Q s˙ ∂ s˙

Constant Pressure

pi ≡


NPT MTK NPT

Hamiltonian of the extended system (N particles + s) HNose = .

N ∑ i

CP-MD

p2i ps2 − (3N + 1)kTeq ln(s) − U({r}) + 2Q 2mi s2

1.28


MD


28 / 58

Introduction

Canonical Ensemble


. Nosé equations of motion . virtual variables:p, r, t

.


NVT Nosé - Hoover

dri ∂H pi = = dt dpi mi s 2 dpi ∂H U({r}) = =− dt dri ∂ri ds ∂H ps = = dt dps Q ) ( N ∑ p2 dps ∂H 1 i = = − (3N + 1)kTeq 2 dt ds s mi s


NPT MTK NPT

CP-MD

i

1.29


MD


29 / 58

Introduction

Canonical Ensemble

The model

. real variables . r and s correspond to the real variables, while ⃗π = p/s , ′ .ps = ps /s and τ = t/s

The simulation

. equations of motion .

MTK


NVT Nosé - Hoover

Constant Pressure Andersen MTK

.

dri dri pi ⃗πi =s = = dτ dt mi s mi d⃗πi dpi /s dpi 1 ds =s = − pi dτ dt dt s dt ds ds sps′ =s = dτ dt Q ) ( N ′ ∑ ⃗π 2 1 dsps /Q s dps i − (3N + 1)kTeq = = dτ Q dt mi Q


NPT MTK NPT

CP-MD

i

1.30


MD


30 / 58

Introduction

Canonical Ensemble


. Constant of the motion in real variables .


NVT

HNose

N ∑ ⃗πi2 = + U({r}) 2mi

Nosé - Hoover


i

.

Andersen

(sps′ )2 + + (3N + 1)kTeq ln s 2Q


NPT

• transformation to real variables is not canonical

MTK NPT

CP-MD

• HNose It is not an Hamiltonian • implementation in real variables • eqm are not easy to be implemented in MD

1.31


MD


31 / 58

Introduction

Canonical Ensemble

The model

. Nosé - Hoover equations . Hoover Phys.Rev.A31 (1985)1695: non canonical change of variables in Nosé eqm ds ⃗πi = psi ,dτ = dts , 1s dτ = dη dt and ps = pη and posing the number of dof to 3N

.

pi r˙i = m pη p˙ i = −∇i U({r}) − pi Q pη η˙ = Q N ∑ p2i p˙ η = − 3NkTeq mi


NVT Nosé - Hoover


NPT MTK NPT

CP-MD

i

1.32


MD


32 / 58

Canonical Ensemble . Nosé - Hoover . • η acts as a friction term • p˙ η twice the difference of kinetic energy and its canonical average


NVT Nosé - Hoover

MTK

• It is a non Hamiltonian system

Constant Pressure Andersen

Conserved Energy

MTK PR Nosé Klein

pη2 H(r, p, η, pη ) = H(r, p) + + 3NkTeq 2Q

Wentzcovitch

NPT MTK NPT

The real ∑Hamiltonian is ηH(r, ∑Np). When N F = 0 also e i i i Fi is constant and the distribution function of the momenta is wrong

.

CP-MD

p2 1 f (p) ̸= √ e− 2mkT 2πmkT 1.33


MD


33 / 58

Introduction

Canonical Ensemble


. Nosé - Hoover . 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! .


NVT Nosé - Hoover


. Nosé - Hoover Chain . The f (pη ) must follow a Maxwell-Boltzmann distribution pη must be coupled to a thermostat. The chain of thermostats should be infinite The lenght is chosen finite (M thermostats) .

Wentzcovitch

NPT MTK NPT

CP-MD

1.34


MD


34 / 58

Introduction

Canonical Ensemble . Nosé - Hoover Chain eqm .

.


pi r˙ i = m pη p˙ i = Fi − 1 pi Q1 p ηj η˙j = j = 1, · · · · · · , M Qj   N 2 ∑ pi pη p˙ η1 =  − 3NkT  − 2 pη1 mi Q2 i=1,N [ 2 ] pηj−1 pηj+1 j = 2, · · · · · · , M − 1 p˙ ηj = − kT − pη Qj−1 Qj+1 j [ 2 ] pηM−1 p˙ ηM = − kT QM−1


NVT Nosé - Hoover


NPT MTK NPT

CP-MD

1.35


MD


35 / 58

Introduction

Canonical Ensemble


. Nosé - Hoover Chain . Previous equations cannot be transformed in an Hamiltonian system Optimal choice: Q1 = 3NkT τ 2 , Qj = NkT τ 2 j = 2, · · · , M τ. > 10∆t . Conserved Energy . H = H(r, p) +

NVT Nosé - Hoover


NPT MTK NPT

M p2 ∑ ηj j=1

.


2Qj

+ 3NkT η1 + kT

M ∑

CP-MD

ηj

j=2

1.36


MD


36 / 58

Introduction

Canonical Ensemble


NVT

. Massive Nosé - Hoover 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) 12959 rapid thermalization of a protein in solution .

Nosé - Hoover


NPT MTK NPT

CP-MD

1.37


MD


37 / 58

Introduction

Canonical Ensemble


. Integration of Nosé - Hoover Chain eqm . The Liouville operator can be written as:

NVT Nosé - Hoover


ıL = ıLNHC + ıL1 + ıL2

Andersen MTK PR

.e

ıL∆t

=e

ıLNHC ∆t/2 ıL2 ∆t/2 ıL1 ∆t ıL2 ∆t/2 ıLNHC ∆t/2

e

e

e

e

3

+O(∆t )


NPT

. Velocity Verlet . eıLH ∆t = eıL2 ∆t/2 eıL1 ∆t eıL2 ∆t/2 .

MTK NPT

CP-MD

1.38


MD


38 / 58

Introduction

Canonical Ensemble . Thermostat forces .






G1 = 


N ∑ pi2 − 3NkT  mi

NVT Nosé - Hoover

MTK

i=1,N

Gj = .

pη2j−1 Qj−1

Constant Pressure Andersen

− kT


.

NPT MTK NPT

ıLNHC = −

N ∑ pη i=1

+ .

∑ pηj ∂ ∂ + ∂pi Qj ∂ηj M

1

Q1

M−1 ∑(

pi ·

G j − p ηj

j=1

CP-MD

j=1

pηj+1 Qj+1

)

∂ ∂ + GM ∂pηj ∂pηM 1.39


MD


39 / 58

Introduction

Canonical Ensemble


NVT

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]

Nosé - Hoover


NPT MTK NPT

CP-MD

1.40


MD


40 / 58

Introduction

Canonical Ensemble

The model

. Suzuki-Yoshida . Given a primitive factorization, S(λ), of an operator


NVT

S(λ) = eλA2 /2 eλA2 eλA1 /2 = eλ(A1 +A2 )

Nosé - Hoover

MTK

chosen an even order of the error in the factorization 2s this gives nSY = 5s−1 weigths, wα , such that:

Constant Pressure Andersen MTK PR Nosé Klein

nSY ∑

Wentzcovitch

wα = 1

α=1

NPT MTK NPT

CP-MD

The factorization is eλ(A1 +A2 ) ≈ .

nSY ∏

S(wα λ)

α=1 1.41


MD


41 / 58

Introduction

Canonical Ensemble


. RESPA . Setting λ = ∆t/2


NVT Nosé - Hoover

MTK

e

ıLNHC ∆t/2

≈

nSY ∏

Constant Pressure

S(wα ∆t/2)

Andersen MTK PR

α=1


the operator is to be applied n times with a time step wα ∆t 2n [n ( )]n SY ∏ ∆t ıLNHC ∆t/2 e ≈ S wα 2n α=1 .

NPT MTK NPT

CP-MD

1.42


MD


42 / 58

Introduction

Canonical Ensemble


. Factorization of eıLNHC ∆t/2 . The choice is not unique ∂ A new kind of operator ecx ∂x Given a function p, the action of the operator is [∞ ] ∑ c k ( ∂ )k ∂ cp ∂p e p p= p k! ∂p =p

k=0 ∞ ∑

k =0


NVT Nosé - Hoover


NPT

ck = pec k!

MTK NPT

CP-MD

c .Given a function f (p), the results is f (pe )

Terms of this kind will act scaling the momentum of the thermal bath. 1.43


MD


43 / 58

Isoenthalpic-Isobaric Ensemble . Andersen . H.C.Andersen, J.Chem. Phys. 72 (1980) 2384 First attempt to perform MD simulations at costant P The volume,V , is treated as a dynamical variable. Cartesian coordinates {ri } are replaced by scaled coordinates {⃗ ρi }: ρ ⃗i = ri V −1/3


NVT Nosé - Hoover


Postulated Lagrangian:

Wentzcovitch

NPT

N N ∑ ∑ ⃗˙ i · ρ ⃗˙ i − ⃗˙ V , V˙ ) = 1 mV 2/3 L({⃗ ρ}, {ρ}, ρ U(V 1/3 ρij ) 2 i=1

MTK NPT

CP-MD

ii

1 ⃗2 + Π + Pext V 2MP

NVT Nosé - Hoover


. Andersen real variables . Conserved Energy H = H0 (r, p) +

NPT MTK NPT

CP-MD

pV2 + Pext V 2MP

with pV = MP V˙ . 1.45


MD


45 / 58

Introduction

Isoenthalpic-Isobaric Ensemble . MTK eqm . Martyna, Tobias, Klein J.Chem.Phys. 101(1994)4177 1 V˙ a . variable ϵ = 3 ln(V /V0 ) with momentum pϵ = MP 3V


NVT Nosé - Hoover


pi pϵ + ri mi MP ) ( pϵ 3 p˙ i = Fi − pi 1 + MP (3N − Nconstr ) 3Vpϵ V˙ = MP N 3 ∑ p2i p˙ ϵ = 3V (P int − Pext ) + Ndof mi

Andersen

r˙ i =


NPT MTK NPT

CP-MD

i=1

int ] and P int = with P int = 13 Tr [Pαβ αβ G. Cardini, R. Chelli (U.Firenze)

1 V

∑N [ piα piβ i=1 MD

mi

] + Fiα riβ 1.46


46 / 58

Isoenthalpic-Isobaric Ensemble . Parrinello - Rahman . J.Appl.Phys. 52(1981)7182 Parrinello and Rahaman same estended Lagrangian of Andersen but with variable cell shape transformation matrix from scaled,s, to cartesian,r   ax bx cx H = ay by cy  az bz cz r = Hs real distances by metric matrix G = Ht H √


NVT Nosé - Hoover


NPT MTK NPT

CP-MD

V = det H, cell sides a = ax2 + ay2 + az2 .... 9 elements of the matrix are used as dynamical variables. 3 sides and 3 angles of the cell. .3 Euler angles!! G. Cardini, R. Chelli (U.Firenze)

MD


1.47

47 / 58

Isoenthalpic-Isobaric Ensemble . Molecular case . Nosé and Klein Mol. Phys. 50 (1983) 1055 No rotating simulation box   H11 H12 H13 H =  0 H22 H23  0 0 H33


NVT Nosé - Hoover


H11 = a

NPT

H12 = b cos(γ)

CP-MD

MTK NPT

H13 = c cos(β)

.

H22 = b sin(γ) cos(α) − cos(β) cos(γ) H23 = c sin(γ) √ 2 − H2 H33 = c 2 − H13 23 G. Cardini, R. Chelli (U.Firenze)

MD

1.48


48 / 58

Introduction

Isoenthalpic-Isobaric Ensemble


. Molecular case . Center of mass scaling Small molecules center of mass variables: ⃗α Rα ; Pα ; F internal Pmolec

.

NVT Nosé - Hoover


] Molec [ 1 ∑ P2α ⃗ = + Rα · Fα 3V Mα

NPT MTK NPT

CP-MD

α=1

1.49


MD


49 / 58

Introduction

Isobaric Ensemble


NVT Nosé - Hoover

. Wentzcovitch . Phys. Rev. B44 (1991) 2358 Dynamical variable Stress Tensor Very useful to optimize crystal structure .


NPT MTK NPT

CP-MD

1.50


MD


50 / 58

Introduction

NPT


. es Molecules . L =

N ∑ α=1


{

n ∑ i=1

miα ⃗ 2 s2 σ˙ iα 2

}

Mα 2 ⃗ t ⃗ s (H ρ˙ α ) (H ρ˙ α ) + 2

NVT Nosé - Hoover


WP ˙ − U({⃗σiα + H⃗ ρα }) + Tr (H˙ t H) 2 Ws 2 s˙ − Pext det(H) − 3NkText ln s + 2


NPT MTK NPT

CP-MD

.NPT Lagrangian with a simple Nosé Nosé-Hoover chain separated thermostats for atoms and barostat

1.51


MD


51 / 58

Introduction

NPT . MTK factorization . Tuckerman et al. J. Phys. A39 (2006) 5629 factorization scheme for an atomic system


NVT

e

ıL∆t

=

eıLNHC−baro ∆t/2 eıLNHC−part ∆t/2 eıLg,2 ∆t/2 eıL2 ∆t/2 eıLg,1 ∆t eıL1 ∆t eıL2 ∆t/2

Nosé - Hoover


.

eıLg,2 ∆t/2 eıLNHC−part ∆t/2 eıLNHC−baro ∆t/2


NPT

˙ −1 pg = Wp HH ] N [ ∑ pg pi ıL1 = + ri mi Wp i=1 ( ) ] N [ ∑ ∂ 1 Tr[pg ] ıL2 = Fi − pg Wp + I pi · N f WP ∂pi

MTK NPT

CP-MD

i=1

1.52


MD


52 / 58

Introduction

NPT


ıLg,1

pg H ∂ = Wp ∂H [

ıLg,1 = det[H](Pint


]

N 1 ∑ p2i ∂ − IPext ) + I Nf mi ∂pg i=1

ıLNHC−part = −

N ∑ i=1

p η1 ∂ pi · + Q1 ∂pi

M ∑ j=1

NVT Nosé - Hoover


p ηj ∂ Qj ∂ηj


NPT

) M−1 ∑( pηj+1 ∂ ∂ Gj − pηj + GM Qj+1 ∂pηj ∂pηM

MTK NPT

CP-MD

j=1

ıLNHC−baro = same as ıLNHC−part with pg

1.53


MD


53 / 58

Introduction

NPT


. pseudo masses . For T of particles:


NVT Nosé - Hoover

Q1 = 3kT τp2

Qj = kT τp2


For T of barostat


Q1 =

kT τb2

Qj =

kT τb2

NPT MTK NPT

CP-MD

For pressure WP = (3N + 3)kT τb2 Martyna 1992, 1996 .

1.54


MD


54 / 58

Introduction

First Principles Molecular Dynamics: CP-MD


Unified Approach for Molecular Dynamics and Density-Functional Theory R. Car and M. Parrinello Phys. Rev. Lett. 55 (1985) 2471 . CP-MD extended Lagrangian .


NVT Nosé - Hoover


L=

occ.orb. ∑ ∫ i

+ .

∑

(∫ λij

µi ∥ψ˙ i (r)∥2 dr + ψi∗ (r)ψi (r)dr

1∑ 2

MTK PR

˙ 2 − E[{ψi }, Rα ] Mα R α


NPT

α

)

MTK NPT

− δij

CP-MD

ij

An intertial factor µi , pseudomass (mass ∗ lenght 2 ), is associated to the electronic degree of freedom. 1.55


MD


55 / 58

Introduction



. CP-MD: equations of motion . Assumption: the system is on the BO surface µψ¨i = −

occ.orb. ∑ δE + λij ψj i = 1, ..., occ.orb δ ψ˙ i j=1


NVT Nosé - Hoover

MTK

(1)


¨ I = −∇R E I = 1, .., Atoms MI R I

NPT

(2)

MTK NPT

CP-MD

Hellman-Feynman .

⟨ ⟩ −∇RI E({RI }) = − Ψ0 | ∇RI H | Ψ0

1.56


MD


56 / 58

Introduction


The model The simulation Integrators Velocity Verlet

. CP-MD: trajectories . The trajectories generated by CP equations does not corresponds to the true ones unless

Multiple time step

NVT Nosé - Hoover


E[{ψi }, {RI }]


is . in the minimum respect {ψi } at each time step This is obtained by choosing the value of µ to obtain a decoupling between electronic and nuclear degree of freedom

NPT MTK NPT

CP-MD

1.57


MD


57 / 58

Introduction

Rare events

The model

. Metadynamics . Laio, Parrinello PNAS 99(2002) 12562 Iannuzzi, Laio, Parrinello PRL 90(2003) 238302 .

The simulation

. Estended Lagrangian . collective variables sα

MTK


NVT Nosé - Hoover

Constant Pressure Andersen MTK PR

L

MTD

=L+

∑1 α

2

Mα s˙ α2 −

∑1 α

2

Nosé Klein

kα [Sα − sα ] − V (t, s) 2

Wentzcovitch

NPT MTK NPT

CP-MD

kα 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 .50-100 MD steps 1.58


MD


58 / 58

Newtonian and extended Lagrangian dynamics

Newtonian and extended Lagrangian dynamics

Suggest Documents

Extended Lagrangian free energy molecular dynamics

Lagrangian formulation of Newtonian cosmology

Second post-Newtonian Lagrangian dynamics of spinning compact ...

An extended Lagrangian formalism

1 Newtonian and Lagrangian Mechanics of a ...

BUBBLE DYNAMICS AND CAVITATION IN NON-NEWTONIAN ...

Newtonian limit of Extended Theories of Gravity

Lagrangian submanifolds and dynamics on Lie algebroids

REDUCED DYNAMICS AND LAGRANGIAN SUBMANIFOLDS OF ...

Extended Lagrangian approach for the defocusing

Displacing Lagrangian toric fibers by extended probes

Geometric Structures, Lobe Dynamics, and Lagrangian ... - CiteSeerX

The Magellanic Stream in Modified Newtonian Dynamics

Modified Newtonian Dynamics, an Introductory Review - arXiv

Spherical Collapse in Modified Newtonian Dynamics (MOND)

Comparing different realizations of modified Newtonian dynamics ...

N-BODY SIMULATIONS IN MODIFIED NEWTONIAN DYNAMICS

Spherical Collapse in Modified Newtonian Dynamics (MOND)

dissipationless collapses in modified newtonian dynamics - IOPscience

[PDF] Download Newtonian Dynamics Audiobooks - Google Sites

ON DYNAMICS OF LAGRANGIAN TRAJECTORIES FOR HAMILTON

Eulerian vs. Lagrangian analyses of pedestrian dynamics

Lagrangian dynamics approach for the derivation of

Comparison of Newtonian and Non-Newtonian ...