An Introduction to Advanced Molecular Dynamics Techniques

7/25/2019 An Introduction to Advanced Molecular Dynamics Techniques

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Marco MasiaDipartimento di Chimica - Universit di Sassari

http://physchem.uniss.it/marco.masia

[email protected]


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Molecular Dynamics - Bibliography


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Molecular Dynamics - Bibliography


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Molecular Dynamics

In Molecular Dynamics the Lagrangian and the Hamiltonian have the following form:


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Molecular Dynamics


L(rN,rN) =N

i=1

mi

2 ri ri

i


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Molecular Dynamics


L(rN,rN) =N

i=1

mi

2 ri ri

i


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Molecular Dynamics


L(rN,rN) =N

i=1

mi

2 ri ri

i


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics

Some Questions:


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Molecular Dynamics


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Molecular Dynamics

The equations of motion are propagated in time through the classical propagator:

x(t) = eiLtx(0)


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Molecular Dynamics


x(t) = eiLtx(0)

If the Liouville operator could be written as a sum of two operators, the classical

propagator could be rewritten making use of the Trotter theorem:

exp(iLt) = exp[(iL1+ iL2)t] = limP

exp

iL2t

2P

exp

iL1t

P

exp

iL2t

2P

P


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Molecular Dynamics


x(t) = eiLtx(0)




exp

iL2t

2P

exp

iL1t

P

exp

iL2t

2P

P

For finitePwe define and we approximate the classical propagator as:t = t/P


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Molecular Dynamics


x(t) = eiLtx(0)




exp

iL2t

2P

exp

iL1t

P

exp

iL2t

2P

P


exp(iLt) exp(iL2t/2) exp(iL1t) exp(iL2t/2) +O(t3)


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Molecular Dynamics


x(t) = eiLtx(0)




exp

iL2t

2P

exp

iL1t

P

exp

iL2t

2P

P


exp(iLt) exp(iL2t/2) exp(iL1t) exp(iL2t/2) +O(t3)

exp(iLPt) P

k=1

exp(iL2t/2) exp(iL1t) exp(iL2t/2) +O(tt2)


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Molecular Dynamics

Given the Liouville operator:

iL = {. . . ,H} =Ni=1

H

pi

ri

H

ri

pi

=

Ni=1

pi

mi

ri+ Fi

pi


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Molecular Dynamics


iL = {. . . ,H} =Ni=1

H

pi

ri

H

ri

pi

=

Ni=1

pi

mi

ri+ Fi

pi

Lets consider the following partition:

iL2 =

N

i=1

Fi

piiL1 =

N

i=1

pi

mi

ri


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Molecular Dynamics


iL = {. . . ,H} =Ni=1

H

pi

ri

H

ri

pi

=

Ni=1

pi

mi

ri+ Fi

pi


iL2 =

N

i=1

Fi

piiL1 =

N

i=1

pi

mi

ri

exp(iL1t)

ri ri+ t(pi/mi)


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Molecular Dynamics


iL = {. . . ,H} =Ni=1

H

pi

ri

H

ri

pi

=

Ni=1

pi

mi

ri+ Fi

pi


iL2 =

N

i=1

Fi

piiL1 =

N

i=1

pi

mi

ri

exp(iL2t/2)

pi pi+ (t/2)Fi(r)

exp(iL1t)

ri ri+ t(pi/mi)


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Molecular Dynamics

The action of the operator on the full set of momenta and positionscan be evaluated analytically, yielding the following approximate evolution equations:exp(iLPt

)


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Molecular Dynamics

The action of the operator on the full set of momenta and positionscan be evaluated analytically, yielding the following approximate evolution equations:exp(iLPt

)

ri(t) = ri(0) + vi(0)t+t

2

2miFi(0)

vi(t) = vi(0) + t2

2mi[Fi(0) + Fi(t)]

These equations constitute

the so-called velocity Verletintegrator. According to theapproximate propagator

expression, we can look atthe integration procedure as

if it was a set of threesequential updates:


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Molecular Dynamics

The action of the operator on the full set of momenta and positionscan be evaluated analytically, yielding the following approximate evolution equations:

exp(iLPt)

ri(t) = ri(0) + vi(0)t+t

2

2miFi(0)

vi(t) = vi(0) + t2

2mi[Fi(0) + Fi(t)]

These equations constitute

the so-called velocity Verletintegrator. According to theapproximate propagator

expression, we can look atthe integration procedure as

if it was a set of threesequential updates:

vi vi +Fit/2mi

vi vi +Fit/2mi

ri ri + vi t

Fi


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Molecular Dynamics

Some Remarks:


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Molecular Dynamics

Some Remarks:


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Molecular Dynamics

Some Remarks:


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Molecular Dynamics

Some Remarks:


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Molecular Dynamics

Some Remarks:


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Molecular Dynamics

In the case of non-Hamiltonian systems it has been seen that the Lioville operatorcould be split in two parts:

iL=

N

i=1

xi

xi=

N

i=1

i(x)

xi= iLH +iLnH

Since the Trotter theorem is easily extended to the exponential of the sum of more

than two operators, we could construct the propagator as follows:

exp(iLt) exp(iLnHt/2)exp(iL(2)Ht/2)exp(iL

(1)Ht)exp(iL

(2)Ht/2)exp(iLnHt/2)

iLnH


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Molecular Dynamics

Many different integration schemes can be derived using the direct translation

technique with different partitions of the propagator. All of them (usually called

Verlet-like schemes) guarantee time reversibility and energy conservation at long

times. Only for pedagogical purposes we introduce here the Verlet Leaprogscheme:


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Molecular Dynamics





The velocities and

positions are updated

at different times,

which hinders the

definition of the total

energy!

ri(

t) =ri

(0) +vi

(

t/2)

t+

t2

miFi

(0)

vi(t/2) = vi(t/2)t+t2

miFi(0)


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Molecular Dynamics





The velocities and

positions are updated

at different times,

which hinders the

definition of the total

energy!

ri(

t) =ri

(0) +vi

(

t/2)

t+

t2

miFi

(0)

vi(t/2) = vi(t/2)t+t2

miFi(0)

ri ri + vit

Fi

t

t+t/2 vi vi +Fit/mi


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Molecular Dynamics


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Molecular Dynamics

lim

A =A


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Molecular Dynamics

In Hamiltonian systems the ensemble average is generated according to a microcanonicaldistribution. In order to generate trajectories which sample other statistical mechanicalensembles Andersen introduced the use of extended Lagrangians.

lim

A =A


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Molecular Dynamics

In Hamiltonian systems the ensemble average is generated according to a microcanonicaldistribution. In order to generate trajectories which sample other statistical mechanical

ensembles Andersen introduced the use of extended Lagrangians.

lim

A =A


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Molecular Dynamics



lim

A =A

L(rN,rN) =

N

i=1

mi

2 ri ri

i


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Molecular Dynamics



lim

A =A

L(rN,rN, V, V) =N

i=1

mi

2 ri ri

i


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Molecular Dynamics



lim

A =A

L(rN,rN, V, V) =N

i=1

mi

2 ri ri

i


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Molecular Dynamics



lim

A =A

L(rN,rN, V, V) =N

i=1

mi

2 ri ri

i


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Molecular Dynamics

L(rN

,rN

, V, V) =

N

i=1

mi

2 ri

ri

i


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Molecular Dynamics

L(rN

,rN

, V, V) =

N

i=1

mi

2 ri

ri

i


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Molecular Dynamics

L(rN

,rN

, V, V) =

N

i=1

mi

2 ri

ri

i


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Molecular Dynamics

L(rN

,rN

, V, V) =

N

i=1

mi

2 ri

ri

i


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Molecular Dynamics

L(rN

,rN

, V, V) =

N

i=1

mi

2 ri

ri

i


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Molecular Dynamics

LNH= L(rN,rN, s,s) =

N

i=1

mi

2 s2ri ri

i


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Molecular Dynamics


N

i=1

mi

2 s2ri ri

i


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Molecular Dynamics


N

i=1

mi

2 s2ri ri

i


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Molecular Dynamics


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Molecular Dynamics

dpMdt

=GM

dri

dt=

pi

mi

dpidt =

N

j=i

riju

(rij)

p1

1pi

dk

dt =

pkk

k = 1, . . . ,M

dpkdt

=Gk pk+1

k+1pk k = 1, . . . ,M 1

G1 =

N

i=1

pi pi

mi dNkBT Gk =

pk1

k1 kBT k = 2, . . . , M

1 = dNkBT2 k =kBT

2 k = 2, . . . ,M


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Molecular Dynamics

!

! !"

! !

!" # $#

!" # #!%#

!& !#!

!$% $

%$

&

" !#! !!

!$% $#%#$& !$

%

!$ &"

! #$%&' ()(*('+ , #$%&' ()(*('+- . #$%&' ()(*('+-


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Molecular Dynamics


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Molecular Dynamics

iLNHC = iLr + iLv + iLC


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Molecular Dynamics


iLr =

N

i=1

vi ri

iLv =

N

i=1

Fi

miv

i


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Molecular Dynamics


iLr =

N

i=1

vi ri

iLv =

N

i=1

Fi

miv

i

iLC =

M

k=1

vk

k

M

i=1

vi vi +

M1

k=1

Gk vkvk+1

vk

+GM

vM


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Molecular Dynamics


iLr =

N

i=1

vi ri

iLv =

N

i=1

Fi

miv

i

iLC =

M

k=1

vk

k

M

i=1

vi vi +

M1

k=1

Gk vkvk+1

vk

+GM

vM

eiLt

= eiLCt/2

eiLvt/2

eiLrt

eiLvt/2

eiLCt/2


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Molecular Dynamics

iLC =

5

j=1

iLj

C


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Molecular Dynamics

iLC =

5

j=1

iLj

C

eiLCt

= eiL5

Ct/4

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL5

Ct/4

eiL

Ct/2

eiL1Ct/2


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Molecular Dynamics

iLC =

5

j=1

iLj

C

eiLCt

= eiL5

Ct/4

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL5

Ct/4

eiL

Ct/2

eiL1Ct/2

eiLt

= eiLCt/2

eiLvt/2

eiLrt

eiLvt/2

eiLCt/2


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Molecular Dynamics

iLC =

5

j=1

iLj

C

eiLCt

= eiL5

Ct/4

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL5

Ct/4

eiL

Ct/2

eiL1Ct/2

eiLt

= eiLCt/2

eiLvt/2

eiLrt

eiLvt/2

eiLCt/2


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Molecular Dynamics

iLC =

5

j=1

iLj

C

eiLCt

= eiL5

Ct/4

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL4

Ct/8

eiL3

Ct/4

eiL4

Ct/8

eiL5

Ct/4

eiL

Ct/2

eiL1Ct/2

eiLt

= eiLCt/2

eiLvt/2

eiLrt

eiLvt/2

eiLCt/2


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Molecular Dynamics


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Molecular Dynamics

(x, t) = x=Ni=1

[riri+ pipi] +Mk=1

kk

+pkpk

= dN1

Mk=2

k

d= exp

1+

Mk=2

M

dNpdNrdMpdM


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Molecular Dynamics

(x, t) = x=Ni=1

[riri+ pipi] +Mk=1

kk

+pkpk

= dN1

Mk=2

k

d= exp

1+

Mk=2

M

dNpdNrdMpdM


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Molecular Dynamics


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Molecular Dynamics

U= U0 + Upol

Upol =

i

Ei Pi +

i

Pi Pi

2i


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Molecular Dynamics

Ei = TijPi Pi = iEi

U= U0 + Upol

Upol =

i

Ei Pi +

i

Pi Pi

2i


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Molecular Dynamics

Ei = TijPi Pi = iEi

U= U0 + Upol

Upol =

i

Ei Pi +

i

Pi Pi

2i


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Molecular Dynamics

Ei = TijPi Pi = iEi

U= U0 + Upol

Upol =

i

Ei Pi +

i

Pi Pi

2i


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Molecular Dynamics

Ei = TijPi Pi = iEi

U= U0 + Upol

Upol =

i

Ei Pi +

i

Pi Pi

2i


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Molecular Dynamics

Ei = TijPi Pi = iEi

U= U0 + Upol

Upol =

i

Ei Pi +

i

Pi Pi

2i


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Molecular Dynamics

l l


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Molecular Dynamics

M l l D i


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Molecular Dynamics

L(rN,rN,PM, PM) = 12

N

i=1

miri ri+12

M

i=1

iPi Pi U(r

N,P

M)

M l l D i


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Molecular Dynamics


N

i=1

miri ri+12

M

i=1

iPi Pi U(r

N,P

M)

dri

dt=

pi

mi

dpidt

=

N

j=i

rijU(rij)

id2Pi

dt2 =

L

Pi=

Pi

i+Ei

M l l D i


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Molecular Dynamics


N

i=1

miri ri+12

M

i=1

iPi Pi U(r

N,P

M)

dri

dt=

pi

mi

dpidt

=

N

j=i

rijU(rij)

id2Pi

dt2 =

L

Pi=

Pi

i+Ei

M l l D i


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Molecular DynamicsSome Remarks:

M l l D i


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M l l D i


85/142


M l l D i


86/142


M l l D i


87/142


M l l D i


88/142


M l l D i


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Molecular Dynamics

M l l D i


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Molecular Dynamics

M l l D i


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Molecular Dynamics

M l l D i


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Molecular Dynamics

MoldMnew

Molecular Dynamics


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Molecular Dynamics

MoldMnew

Molecular Dynamics


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Molecular Dynamics

Molecular Dynamics


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Molecular Dynamics

Fi = Fref

i +F

del

i

Fdel

i = F

ref

i Fi

Molecular Dynamics


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Molecular Dynamics

Fi = Fref

i +F

del

i

Fdel

i = F

ref

i Fi

F

del

i

Fref

i

Molecular Dynamics


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Molecular Dynamics

Fi = Fref

i +F

del

i

Fdel

i = F

ref

i Fi

F

del

i

Fref

i

Molecular Dynamics


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Molecular Dynamics

Fi = Fref

i +F

del

i

Fdel

i = F

ref

i Fi

F

del

i

Fref

i

t t

Molecular Dynamics


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Molecular Dynamics

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

iLref=

Ni=1

pi

mi

ri+Frefi

pi

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

iLref=

Ni=1

pi

mi

ri+Frefi

pi

iL

del=

N

i=1

Fdeli

pi

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

iLref=

Ni=1

pi

mi

ri+Frefi

pi

iL

del=

N

i=1

Fdeli

pi

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

iLref=

Ni=1

pi

mi

ri+Frefi

pi

iL

del=

N

i=1

Fdeli

pi

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

iLref=

Ni=1

pi

mi

ri+Frefi

pi

iL

del=

N

i=1

Fdeli

pi

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

iLref=

Ni=1

pi

mi

ri+Frefi

pi

iL

del=

N

i=1

Fdeli

pi

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

Molecular Dynamics


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Molecular Dynamics

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

exp(iLt) = exp

iL

delt

2

exp

iL

ref

2

t

2

exp

iL

ref

1 t exp

iL

ref

2

t

2

n

exp

iL

delt

2

Molecular Dynamics


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Molecular Dynamics

Molecular Dynamics


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Molecular Dynamics

Molecular Dynamics


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Molecular Dynamics

Molecular Dynamics


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Molecular Dynamics

f(r1, r2, . . . , rN, t) = 0;

Molecular Dynamics


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y

f(r1, r2, . . . , rN, t) = 0;

=|ri(t)

rj(t)|2

d2ij = 0

Molecular Dynamics


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y

f(r1, r2, . . . , rN, t) = 0;

L(rN,rN) =1

2

N

i=1

miri ri U(rN)

Nc

=1

(rN)

=|ri(t)

rj(t)|2

d2ij = 0

Molecular Dynamics


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y

dri

dt

= pi

mi

dpidt

=

N

j=i

riju(rij)

Nc

=1

ri=Fi+Gi

Molecular Dynamics


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y

dri

dt

= pi

mi

dpidt

=

N

j=i

riju(rij)

Nc

=1

ri=Fi+Gi

Molecular Dynamics


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y

dri

dt

= pi

mi

dpidt

=

N

j=i

riju(rij)

Nc

=1

ri=Fi+Gi

Molecular Dynamics


119/142

y

dri

dt

= pi

mi

dpidt

=

N

j=i

riju(rij)

Nc

=1

ri=Fi+Gi

t= 0

t

=

N

i=1

1

mi

Fii

N

i=1

1

mi

Nc

=1

ii+

i,j

rirjij = FM+T = 0

Molecular Dynamics


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y

dri

dt

= pi

mi

dpidt

=

N

j=i

riju(rij)

Nc

=1

ri=Fi+Gi

t= 0

t

=

N

i=1

1

mi

Fii

N

i=1

1

mi

Nc

=1

ii+

i,j

rirjij = FM+T = 0

= M1 (F+T)

Molecular Dynamics


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y

dri

dt

= pi

mi

dpidt

=

N

j=i

riju(rij)

Nc

=1

ri=Fi+Gi

t= 0

t

=

N

i=1

1

mi

Fii

N

i=1

1

mi

Nc

=1

ii+

i,j

rirjij = FM+T = 0

= M1 (F+T)

Molecular Dynamics


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yiterative algorithm (SHAKE/RATTLE):

Molecular Dynamics


123/142

yiterative algorithm (SHAKE/RATTLE):

Molecular Dynamics


124/142

yFew words about the method of constraints to computefree energy differences:

Molecular Dynamics


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F(2) F(1) =

2

1

d

H

Molecular Dynamics


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F(2) F(1) =

2

1

d

H

Molecular Dynamics


127/142

Few words about the method of constraints to computefree energy differences:

F(2) F(1) =

2

1

d

H

(rN

) =

(rN

,

rN

) = 0

Molecular Dynamics


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Few words about the method of constraints to computefree energy differences:

F(2) F(1) =

2

1

d

H

(rN

) =

(rN

,

rN

) = 0

F(2) F(1) =

2

1

ddF

d

Molecular Dynamics


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Molecular Dynamics


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F(2) F(1) =

2

1

ddF

d

Molecular Dynamics


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F(2) F(1) =

2

1

ddF

d

dF

d=

Z1/2 [+ kBTG]

Z

1/2

Molecular Dynamics


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F(2) F(1) =

2

1

ddF

d

dF

d=

Z1/2 [+ kBTG]

Z

1/2

Molecular Dynamics


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F(2) F(1) =

2

1

ddF

d

dF

d=

Z1/2 [+ kBTG]

Z

1/2

Z=Ni=1

1

mi

(rN)

ri

G =

1

Z2

Ni,j

1

mimj

(rN)

ri

2(rN)

rirj

(rN)

rj

Molecular Dynamics


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F(2) F(1) =

2

1

ddF

d

dF

d

=

Z1/2 [+ kBTG]

Z

1/2

Z=Ni=1

1

mi

(rN)

ri

G =

1

Z2

Ni,j

1

mimj

(rN)

ri

2(rN)

rirj

(rN)

rj

Molecular Dynamics


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(rN) =|ri rj |

Molecular Dynamics


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(rN) =|ri rj |

Z=

Ni=1

1

mi

(rN)

ri

= constant

G = 1

Z2

Ni,j

1

mimj

(rN)

ri

2(rN)

rirj

(rN)

rj

= 0

Molecular Dynamics


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(rN) =|ri rj |

Z=

Ni=1

1

mi

(rN)

ri

= constant

G = 1

Z2

Ni,j

1

mimj

(rN)

ri

2(rN)

rirj

(rN)

rj

= 0

dF

d=

Z1/2 [+ kBTG]

Z1/2

=

Molecular Dynamics


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(rN) = ni(rN) =

N

k=i

11 + exp [(rik rc)]

Molecular Dynamics


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(rN) = ni(rN) =

N

k=i

11 + exp [(rik rc)]

Molecular Dynamics


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(rN) = ni(rN) =

N

k=i

11 + exp [(rik rc)]

Fig. 4 (a) Mean force of constraint as a function of the coordinationnumber n for the axial (K) and the equatorial (S) site; (b) correspond-ing free energy curves obtained by integration of the mean forces.


141/142

Molecular Dynamics


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Documents

An Introduction to Advanced Molecular Dynamics Techniques