EFFICIENT MOLECULAR DYNAMICS: THERMOSTATS, BAROSTATS AND ...2017_uzh_cp2... · a(q →q 0) =...

EFFICIENT MOLECULAR DYNAMICS:THERMOSTATS, BAROSTATS AND

MULTIPLE TIME STEPS

Venkat Kapil Michele Ceriotti

July 14, 2017

OUTLINE

1 Why molecular dynamics ?

2 How to integrate equations of motion ?3 How to sample a NVT Ensemble ?4 How to sample a NPT Ensemble ?

2 venkat kapil MD

OUTLINE

1 Why molecular dynamics ?2 How to integrate equations of motion ?

3 How to sample a NVT Ensemble ?4 How to sample a NPT Ensemble ?

2 venkat kapil MD

OUTLINE

1 Why molecular dynamics ?2 How to integrate equations of motion ?3 How to sample a NVT Ensemble ?

4 How to sample a NPT Ensemble ?

2 venkat kapil MD

OUTLINE

1 Why molecular dynamics ?2 How to integrate equations of motion ?3 How to sample a NVT Ensemble ?4 How to sample a NPT Ensemble ?

2 venkat kapil MD

WHAT ARE WE CALCULATING ?

P(~p,~q) is the probability distribution at a given thermodynamiccondition.

〈A(~q)〉 =

∫d~q d~p A(~q) P(~p,~q)∫

d~q d~p P(~p,~q)

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6N dimensional integral !

Figure : Integral by quadrature

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6N dimensional integral !

Figure : Integral by importance sampling

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〈A(~q)〉 =

∫d~q d~p A(~q) P(~p,~q)∫

d~q d~p P(~p,~q)= 〈A( ~q(t))〉t

t should be “long enough"!

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Boltzmann’s Ergodic Conjecture (1871)

〈A(~q)〉 =

∫d~q d~p A(~q) P(~p,~q)∫

d~q d~p P(~p,~q)= 〈A( ~q(t))〉t

t should be “long enough"!

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Boltzmann’s Ergodic Conjecture (1871)

IS MY SIMULATION TIME LONG ENOUGH ?

cAA(∆) = 〈A(t) ·A(∆ + t)〉t

Figure : Auto correlation function for the potential and kinetic energy

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IS MY SIMULATION TIME LONG ENOUGH ?

cAA(∆) = 〈A(t) ·A(∆ + t)〉t

Figure : Auto correlation function for the potential and kinetic energy

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WHY NOT MARKOV CHAIN MONTE CARLO ?

Probability of acceptance in MCMC :

a(q→ q′) = min(1, e−β[V(q′)−V(q)])

1 High acceptance vs “making things happen”.2 Curse of large system size.3 Generalized “smart moves”.

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Metropolis et al. JCP (1953)

a(q→ q′) = min(1, e−β[V(q′)−V(q)])

1 High acceptance vs “making things happen”.

2 Curse of large system size.3 Generalized “smart moves”.

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a(q→ q′) = min(1, e−β[V(q′)−V(q)])

1 High acceptance vs “making things happen”.2 Curse of large system size.

3 Generalized “smart moves”.

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a(q→ q′) = min(1, e−β[V(q′)−V(q)])

1 High acceptance vs “making things happen”.2 Curse of large system size.3 Generalized “smart moves”.

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MOLECULAR DYNAMICS

H =3N∑i=0

+ V(q1, .., q3N); ~p = −∂V∂~q

& ~q =~p

Figure : Dynamics conserves energy

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Rahman PR (1964)

HOW TO INTEGRATE EQUATIONS OF MOTION?

= [~q · ∂∂~q

+ ~p · ∂∂~p

~m· ∂∂~q− ∂V

∂~q· ∂∂~p

= iLH ~x

~x = iLH ~x =⇒ ~x(t) = eiLHt ~x(0)

iLH =~p

~m· ∂∂~q− ∂V

∂~q· ∂∂~p

= iLq + iLp

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Phase space vector :~x = (p1, ..,p3N, q1, .., q3N).

How to evolve ~p & ~q ?

~x(t) = eiLHt ~x(0)

The classical propagator!

~x(t) = eiLHt ~x(0)

= [eiLHtM ]M ~x(0)

= [eiLq∆t+iLp∆t]M ~x(0)

≈ [eiLp∆t/2 · eiLq∆t · eiLp∆t/2]M ~x(0)

eτ(A+B) = [e∆τ/2A · e∆τB · e∆τ/2A]M +O(∆τ−3) ∆τ = τ/M

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Tuckerman et al. JCP (1992)

~x(t) = eiLHt ~x(0)

= [eiLHtM ]M ~x(0)

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~x(t) = eiLHt ~x(0)

= [eiLHtM ]M ~x(0)

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Tuckerman et al. JCP (1992), Trotter PAMS (1959)

~x(∆t) = [eiLp∆t/2 · eiLq∆t · eiLp∆t/2] ~x(0) = ?

Given that:

eiLq∆t = e+ ~p~m∆t· ∂∂~q

eiLp∆t = e−∂V∂~q ∆t· ∂∂~p

ec∂∂x f(x, y) = f(x + c, y)

~p→ ~p− ∂V

∂~q ∆t/2

~q→ ~q + ~p~m∆t

~p→ ~p− ∂V∂~q ∆t/2

The Velocity Verlet algorithm!

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Given that:

ec∂∂x f(x, y) = f(x + c, y)

~p→ ~p− ∂V

∂~q ∆t/2

~q→ ~q + ~p~m∆t

~p→ ~p− ∂V∂~q ∆t/2

9 venkat kapil MD

Given that:

ec∂∂x f(x, y) = f(x + c, y)

~p→ ~p− ∂V

∂~q ∆t/2

~q→ ~q + ~p~m∆t

~p→ ~p− ∂V∂~q ∆t/2

9 venkat kapil MD

Given that:

ec∂∂x f(x, y) = f(x + c, y)

~p→ ~p− ∂V

∂~q ∆t/2

~q→ ~q + ~p~m∆t

~p→ ~p− ∂V∂~q ∆t/2

9 venkat kapil MD

Given that:

ec∂∂x f(x, y) = f(x + c, y)

~p→ ~p− ∂V

∂~q ∆t/2

~q→ ~q + ~p~m∆t

~p→ ~p− ∂V∂~q ∆t/2

9 venkat kapil MD

Given that:

ec∂∂x f(x, y) = f(x + c, y)

~p→ ~p− ∂V

∂~q ∆t/2

~q→ ~q + ~p~m∆t

~p→ ~p− ∂V∂~q ∆t/2

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HOW TO CHOOSE THE TIME STEP?

Figure : Presence of multiple time scales in a system

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HOW TO INTEGRATE WITH MULTIPLE TIME STEPS

Figure : Separation of time scales

H =3N∑i=0

+ Vlr(q1, .., q3N) + Vsr(q1, .., q3N)

iLH =~p

~m· ∂∂~q− ∂Vlr

∂~q· ∂∂~p− ∂Vsr

∂~q· ∂∂~p

= iLq + iLlrp + iLsr

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H =3N∑i=0

+ Vlr(q1, .., q3N) + Vsr(q1, .., q3N)

iLH =~p

~m· ∂∂~q− ∂Vlr

∂~q· ∂∂~p− ∂Vsr

∂~q· ∂∂~p

= iLq + iLlrp + iLsr

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~x(∆t) = eiLH∆t ~x(0)

= ei[iLq+iLsrp +iLlr

p ]∆t ~x(0)

≈ eiLlrp ∆t/2[eiL

srp ∆t+iLq∆t]eiL

lrp ∆t/2 ~x(0)

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p→ p− ∂Vlr

∂~q∆t2

Velocity Verlet for M steps with “short range” forces with ∆t/M.p→ p− ∂Vlr

∂~q∆t2

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MTS: THE REALITY

Figure : Range separation is trivial

[http://www.omnia.md/blog/2014/11/6/how-to-train-your-force-field]

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MTS: THE REALITY

What about the ab inito framework ?

[−−}2

2m∇2 + V(r)] ψ(r) = E ψ(r)

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MTS: THE REALITY

H =3N∑i=0

+ Vsr(q1, .., q3N) + (V(q1, .., q3N)−Vsr(q1, .., q3N))

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Kapil et al. JCP (2016), Marsalek et al. JCP (2016), John et al. PRE (2016)

MTS: THE REALITY

Figure : How to choose the “cheap potential”?

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Kapil et al. JCP (2016), Marsalek et al. JCP (2016), John et al. PRE (2016)

HOW TO INTEGRATE EQUATIONS OF MOTION: WRAPPINGUP

1 Velocity Verlet comes from the classical propagator.

2 Error decreases systematically as (∆t)−2.3 Further decomposition leads to a MTS integrator.4 Time-reversible and symplectic.

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1 Velocity Verlet comes from the classical propagator.2 Error decreases systematically as (∆t)−2.

3 Further decomposition leads to a MTS integrator.4 Time-reversible and symplectic.

14 venkat kapil MD

1 Velocity Verlet comes from the classical propagator.2 Error decreases systematically as (∆t)−2.3 Further decomposition leads to a MTS integrator.

4 Time-reversible and symplectic.

14 venkat kapil MD

1 Velocity Verlet comes from the classical propagator.2 Error decreases systematically as (∆t)−2.3 Further decomposition leads to a MTS integrator.4 Time-reversible and symplectic.

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MOLECULAR DYNAMICS : AT CONSTANT TEMPERATURE

How to sample a NVT ensemble for system given by the Hamiltonian ?

H(~p,~q) =3N∑i=0

+ V(q1, .., q3N)

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Generate (~p,~q) such that:

P(~p,~q) =e−βH(~p,~q)∫

d~pd~qe−βH(~p,~q)

Do Hamilton’s equations of motion conserve P(~p,~q) ?

iLH P(~p,~q) = 0

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Generate (~p,~q) such that:

P(~p,~q) =e−βH(~p,~q)∫

d~pd~qe−βH(~p,~q)

Do Hamilton’s equations of motion conserve P(~p,~q) ?

iLH P(~p,~q) = 0

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Figure : A problem of ergodicity

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Figure : Andersen’s thermostat

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Andersen JCP (1980)

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Andersen JCP (1980)

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Andersen JCP (1980)

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Andersen JCP (1980)

DETERMINISTIC THERMOSTATS

Nose- Hoover thermostat:

~q =~p

~m; ~p = −∂V

∂~q− ~pps

Q; ps = ~p · ~p

~m; s =

1 Not ergodic. Must use chains.2 Not rotationally invariant.3 Integrating equations of motion is not pretty.

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Nosé JCP (1984), Hoover PRA (1985)

~q =~p

~m; ~p = −∂V

∂~q− ~pps

Q; ps = ~p · ~p

~m; s =

1 Not ergodic. Must use chains.

2 Not rotationally invariant.3 Integrating equations of motion is not pretty.

16 venkat kapil MD

~q =~p

~m; ~p = −∂V

∂~q− ~pps

Q; ps = ~p · ~p

~m; s =

1 Not ergodic. Must use chains.2 Not rotationally invariant.

3 Integrating equations of motion is not pretty.

16 venkat kapil MD

~q =~p

~m; ~p = −∂V

∂~q− ~pps

Q; ps = ~p · ~p

~m; s =

1 Not ergodic. Must use chains.2 Not rotationally invariant.3 Integrating equations of motion is not pretty.

16 venkat kapil MD

STOCHASTIC THERMOSTATS

Figure : A white noise Langevin thermostat

Langevin thermostat:

~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

1 Ergodic2 Linear equations3 Integration very stable and easy.

iL = iLγ + iLH; iLγ P(~p,~q) = 0

eiL∆t ≈ eiLγ∆t/2 eiLH∆t eiLγ∆t/2

H = ∆H + ∆K

∆H = Change in total energy during the Hamiltonian step

∆K = Change in kinetic energy during the thermostat step

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Schneider et al. PRB (1978), Bussi et al. JCP (1992)

~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

1 Ergodic

2 Linear equations3 Integration very stable and easy.

H = ∆H + ∆K

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~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

1 Ergodic2 Linear equations

3 Integration very stable and easy.

H = ∆H + ∆K

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~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

H = ∆H + ∆K

17 venkat kapil MD

~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

H = ∆H + ∆K

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~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

H = ∆H + ∆K

17 venkat kapil MD

~q =~p

~m; ~p = −∂V

∂~q− γ~p + ~m1/2

√2γβ−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = δ(t)

H = ∆H + ∆K

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SAMPLING EFFICIENCY

Can we measure how efficient a Langevin thermostat is ?

Figure : Under damped, optimally damped and over damed regimes

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SAMPLING EFFICIENCY

Can we measure how efficient a Langevin thermostat is ?

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Figure : Under damped, optimally damped and over damed regimes

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A GENERALIZED LANGEVIN EQUATION

~q =~p

~p = −∂V∂~q−∫

ds K(s) ~p(t− s) + ~m1/2√2β−1~ξ; 〈~ξ(t) · ~ξ(0)〉 = H(t)

Figure : A generalized Langevin equation (GLE)thermostat

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Zwanzig, Nonequilibrium statistical mechanics (2001)

q =p(p

(−V′(q)

)−(app aTpap A

(bpp bTpbp B

K(t) and H(t) can be expressed in terms of the drift and thediffusion matrix.

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Zwanzig, Nonequilibrium statistical mechanics (2001)

Sampling efficiency over a wide frequency range?

Figure : Optimizing a merit function of sampling efficiency

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Ceriotti et al. JCTC (2010)

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Exciting a narrow range of frequencies?

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Dettori et al. JCTC (2017)

Exciting a narrow range of frequencies?

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Dettori et al. JCTC (2017)

WRAPPING UP

1 System + bath gives canonical sampling.

2 Stochastic modelling of a bath.3 Propagation, ergodicity and conserved quantity X.4 GLE gives both optimal and selective sampling.

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WRAPPING UP

1 System + bath gives canonical sampling.2 Stochastic modelling of a bath.

3 Propagation, ergodicity and conserved quantity X.4 GLE gives both optimal and selective sampling.

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WRAPPING UP

1 System + bath gives canonical sampling.2 Stochastic modelling of a bath.3 Propagation, ergodicity and conserved quantity X.

4 GLE gives both optimal and selective sampling.

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WRAPPING UP

1 System + bath gives canonical sampling.2 Stochastic modelling of a bath.3 Propagation, ergodicity and conserved quantity X.4 GLE gives both optimal and selective sampling.

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WRAPPING UP

1 System + bath gives canonical sampling.2 Stochastic modelling of a bath.3 Propagation, ergodicity and conserved quantity X.4 GLE gives both optimal and selective sampling.

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MOLECULAR DYNAMICS : AT CONSTANT PRESSURE

Figure : Sampling at constant volume

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Andersen JCP (1980), Parinello et al. JAP (1981), Martyna JCP (1994), Bussi JCP (2009)

Figure : Sampling at constant pressure

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~q =~p

~m+ α~q

~p = −∂V∂~q− α~p

V = 3Vαα = 3[V (Pint − Pext) + 2β−1]µ−1

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iL = iLγ + iLH; iLγ PNPT(~p,~q) = 0; iLH PNPT(~p,~q) = 0

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TAKE HOME

1 Molecular Dynamics vs Monte Carlo.

2 Louiville formulation gives robust integrators.3 Canonical sampling can be achieved by stochastic modelling.4 Density fluctuations, changes in cell, isotherms, stress-strain

curves can be computed by sampling the NPT ensemble.

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TAKE HOME

1 Molecular Dynamics vs Monte Carlo.2 Louiville formulation gives robust integrators.

3 Canonical sampling can be achieved by stochastic modelling.4 Density fluctuations, changes in cell, isotherms, stress-strain

24 venkat kapil MD

TAKE HOME

1 Molecular Dynamics vs Monte Carlo.2 Louiville formulation gives robust integrators.3 Canonical sampling can be achieved by stochastic modelling.

4 Density fluctuations, changes in cell, isotherms, stress-straincurves can be computed by sampling the NPT ensemble.

24 venkat kapil MD

TAKE HOME

1 Molecular Dynamics vs Monte Carlo.2 Louiville formulation gives robust integrators.3 Canonical sampling can be achieved by stochastic modelling.4 Density fluctuations, changes in cell, isotherms, stress-strain

24 venkat kapil MD

EFFICIENT MOLECULAR DYNAMICS: THERMOSTATS, BAROSTATS AND ...2017_uzh_cp2... · a(q →q 0) =...

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