Monte-Carlo simulations of the structure of

complex liquids with various interaction

potentialsAljaž

Godec

Advisers: prof. dr. Janko Jamnik and doc. dr. Franci Merzel

National Institute of Chemistry

Contents

1. Introduction

2. Statistical mechanics of complex liquids

3. Spherical multipole expansion of the electrostatic interaction energy

4. Monte-Carlo simulations of ensembles of anisotropic particles

5. How to present the results of MC simulations?

6. Conclusions and considerations for future work

Introduction

What are complex liquids?

simple liquid

anisotropic particles, COMPLEX POTENTIALS

hard spheres, Lennard-Jones

particles-

SIMPLE ISOTROPIC

POTENTIALS

Importance of complex liquids?

≈ρvapour

ρbulk

complex liquid

ΔF= ΔU-TΔS

ΔU = 0 ΔU > 0Hydrophobic interactions

Introduction

≈ρvapour

ρbulk

S

hard sphere in LJ fluid

S

Angular correlations completely ignored!!

Statistical mechanics of complex liquids

Assumption: separable Hamiltonian

(intermolecular interactions have no effect on the quantum states)H=Hclass+Hquant

two sets of independent quantum states (i.e. eigenstates can be taken as a product)

H n E n

cl qun n n with energy

En=Encl+En

qu

The partition function factorisesQ = Q cl Q qu

qu qun i

i

E and

Q ( exp( / ))qu Nqu i

i

kT individual molecular energy

Consequence of the above assumption: the contributions of quantum coordinates to physical properties are independent

of density

classical: centre of mass and the external rotational degrees of freedom

quantum mechanical vibrational and internal rotational degrees of freedom

Statistical mechanics of complex liquids

Probability density for the classical states

The classical Hamiltonian can be split into kinetic and potential energy

( ) exp( ( )) /N N N N N N N NP p H p Z r p r p

d d d d exp( ( ))N N N N N N N NZ p H p r p r p

H=Kt+Kr+U(rNωN)

2

1

/ 2N

t ii

K p m

2

1 , ,

/ 2N

r ii x y z

K J I

0 0

0 0

0 0

xx

yy

zz

I

I

I

I

Iα

( ) ( ) ( ) ( )N N N N N N N NP p P P p P r p p r

In Monte-Carlo calculations we need only the configurational probability density, but

we introduce a new distribution P'(rNpNωNJN)

( ) ( ) NN N N N N N N NP P p Jac r p J r p

1 2

( )...

( )

NN

NN

pJac Jac Jac Jac

J

( )f pJ

Statistical mechanics of complex liquids

new probability density

1

( )sin

( )x y z

p p pJac Jac

J J J

d d d' d ( ) 1N N N N N N N NP r p J r p J

( ) ( ) NN N N N N N N NP P p Jac r p J r p

1( ) ( )(sin ...sin )N N N N N N N NNP P p r p J r p

11( ) '( )(sin ...sin )N N N N N N N N

NP P r p J r p J

it is convinient to introduce a new distribution P(rNpNωNJN)

d d d d ( ) 1N N N N N N N Np P r p r p J

( ) exp( ( )) /N N N N N N N NP H Z r p J r p J

d d d d exp( ( ))N N N N N N N NZ H r p J r p J

We can write

Statistical mechanics of complex liquids

We can now directly factorize( ) ( ) ( ) ( )N N N N N N N NP P P P r p J p J r

t r cZ Z Z Z2

1( ) exp( / 2 ) /

NN

i ti

P p m Z

p

2

1 , ,( ) exp( / 2 ) /

NN

i ri x y z

P J I Z

J

( ) exp( ( )) /N N N NcP U Z r r

2

1

d exp( / 2 )N

Nt i

i

Z p m

p2

1 , ,

d exp( / 2 )N

Nr i

i x y z

Z J I

J

d d exp( ( ))N N N NcZ U r r

1 2( ) ( ) ( )... ( )NNP P P Pp p p p

1 2( ) ( ) ( )... ( )NNP P P PJ J J J

the ps and Js of different molecules are uncorrelated

furthermore

1 1 1 1( ) ( ) ( )... ( )x y zP P p P p P pp

1 1 1 1( ) ( ) ( )... ( )x y zP P J P J P JJ

N N3 3

1Q d d exp( ( ))

! !N N C

N N N N N Nt r t r

ZU

N N

r r

thus we can directly integrateΛt=h/(2πmkT)1/2

Λr=(h/(8π2IxkT)1/2)×

(h/(8π2IykT)1/2)(h/(8π2IzkT)1/2)

Ω=8π2 (4π)

Spherical multipole expansion of the electrostatic interaction energy

electrostatic interaction

1 2

1 2 1 1 2 2

1 2 1 2

1 2 1 212

1( ) ( ) ( ) ( )*m m m

l l l l m l m lml l l m m m

A r r r Y Y Yr

12 6

1 2 12 1212 12

1 2 12 12 12

4q q

Ur r r

a molceule= a distribution of charges (placed in the atomic centres);Atoms have finite sizes and also interact with polarization interactions

electrostatic interaction between two molceules= interaction between two

charge distributionsspherical harmonic expansion of r12

-1=|r+r2-r1|-1

Pair potential energy:

dispersion polarisationexchange repulsion(finite size of atom)

z

y

x

r·

·q1

q2

r1

r2

14( / ) ( ) ( )*

2 1l li

i i lm i lmi i l mi

qq r r Y Y

l

r r

potential of a charge distribution:

Spherical multipole expansion of the electrostatic interaction energy

mth component of the spherical multipole moment of order l :

Q ( )llm i i lm i

i

q r Y

1 2 1 1 2 2

1 2 1 2

112 1 2 1 2/ ( ; )Q Q ( )*l

l l l m l m lml l m m m

u A r C l l l m m m Y

interaction between two charge distributions= ∑(interactions of components of multipole moments of charge

distributions)

z

y

x

r

Introduction of body-fixed coordinate frame:

x’

z’ y’

z

y

x

x’

z’ y’

x’

z’

y’

Ω

' '( ') ( ) ( )lmlm mm lmY D Y

z’’

y’’

x’’

x’

z’

Spherical multipole expansion of the electrostatic interaction energy

z

y

x

y’r·

·q1

q2

r1

r2

xyz: space-fixedx’y’z’ and x’’y’’z’’: body-fixed

Q ( )*Qllm mk lk

n

D

1 2

1 2 1 1 2 2 1 1 2 2

1 2 1 2 1 2

112 1 2 1 2 1 2Q Q / ( ; ) ( )* ( )* ( )*l ll

l l l k l k m k m k lml l k k m m m

u A r C l l l m m m D D Y

calculated only once

What is gained?example: molecule consisting of four charges

17 terms /pair

5 (10) terms /pair

Spherical multipole expansion

TIP5P water model

Relation between multipole moments in the space-fixed and body-fixed

coordinate frames:

1 2 1 1 2 2

1 2 1 2

112 1 2 1 2/ ( ; )Q Q ( )*l

l l l m l m lml l m m m

u A r C l l l m m m Y

12 6

12 12 1 212 12

12 12

4i j ij

q qU

r r r

Monte-Carlo simulations of ensembles of anisotropic particles

What do we do in a MC calculation?2

1

( )dx

x

F f x x2

1

( ( ) / ( )) ( )dx

x

F f x P x P x xP(x)... probability density

function

( )

( ) trials

fF

P

Monte-Carlo: perform a number of trials τ: in each trial choose a random number ζ from P(x) in the interval

(x1,x2)

How to choose P in a way, which allows the function evaluation to be concentrated in the region of space that makes importatnt

contributions to the integral?

Construction of P(x) by Metropolis algorithmgenerates a Markov chain of

states1. outcome of each trial depends only upon the preceding trial

2. each trial belongs to a finite set of possible outcomes

exp( ) / cP U Z

Monte-Carlo simulations of ensembles of anisotropic particles

a state of the system m is characterized by positions and orientations of all moleculesprobability of moving from m to n = πmn

N possible states πmn constitute a N×N matrix, π

each row of π sums to 1

probability that the system is in a particular state is given by the probability vector ρ=(ρ1, ρ2, ρ3,..., ρm, ρn,..., ρN)

(2) (1)ρ ρ π

(3) (1)ρ ρ ππ

probability of the initial state = ρ(1)

lim lim (1) N

N ρ ρ π

equilibrium distribution

Microscopic reversibility (detailed balance):

mn m nm n

if ( )

( / ) if ( )mn n m n m

mnmn n m n m n m

U U

U U

Metropolis: 1mm mnm n

exp( ( ))nn m

m

U U

Monte-Carlo simulations of ensembles of anisotropic particles

How to accept trial moves?Metropolis:

- allways accept if Unew ≤ Uold

- if Unew>Uold choose a

random number ζ from the interval [0,1]

0

1

exp(-βΔU)

Unew-UoldΔUnm

ζ 1

ζ 2×

×allways accept

accept

reject

How to generate trial moves?

1 max

1 max

1 max

0,5

0,5

0,5

i

i

i

x x r

y y r

z z r

translation

rotation max' ( 0,5)

maxcos ' cos ( 0,5) cos

max' ( 0,5)

How many particles should be moved?sampling

efficiency:

2 / ir CPU time

~kT reasonable acceptance 2

2 4

1...

2

0 ( ) ( )

a a bi i ia a b

i i i

ab i

U UU r r r

r r r

f U r

O

U

2 / ( )i abr kT f U

1. N particles, one at a time:

CPU time ~ nN

2 /( ) /i abr CPU kT nf U

2. N particles in one move: 2 /( ) /i abr CPU kT Nnf U

CPU time ~ nN

2 / ( )i abr NkT f U

2 / ( )i abr kT f U sampling efficiency down

by a factor 1/N

Monte-Carlo simulations of ensembles of anisotropic particles

How to represent results (especially angular correlations)?

( ) ! ( ) !exp( ( )) /N N N N N Ncf N P N U Z r r r

we introduce a generic distribution function:

d d ( ) !N N N Nf N r r

! 1( ) d d exp( ( )) / d d ( )

( )! ( )!h h N h N h N N N h N h N N

c

Nf U Z f

N h N h

r r r r r

we further introduce a reduced generic distribution function:

ideal gas:

1 1 2 2( ) ( ) ( )... ( )h hh hf f f f r r r r

1 1 1 1 1 1 1 1 1 1d d ( ) ( ) d d ( )f f V N r r r r r

homogenous isotropic fluid:

1 1( ) /f r

How to present the results of MC simulations?

generally:

pair correlation function:

spherical harmonic expansion of the pair correlation function in a space fixed frame:

( ) ( ) /h h h h h hf g r r

2

12 1 2 3 32

2

1 2 1 22

( 1)( ) d ...d d ...d exp( ( )) /

( ) ( ) ( ) ( )

N NN N c

i j i ji j

N Ng U Z

r r r r

r r r rδ(ω)=δ(φ) δ(cosθ) δ(χ)

1 2

1 1 2 2

1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2( ) ( ; ; ) ( ; ) ( )* ( )* ( )*l lm n m n lm

l l l m m m n n

g g l l l n n r C l l l m m m D D Y r

1 1 2 2

1 2 1 2

1 2 1 2 1 2 1 2 1 2( ) ( ; ) ( ; ) ( ) ( ) ( )*l m l m lml l l m m m

g g l l l r C l l l m m m Y Y Y r

linear molecules:

intermolecular frame ω=0φ :

1 1 2 2( ) ( ) ( )... ( ) ( )h h h hh hf f f f g r r r r rangular correlation function,

g(rhωh) :

1 2

1 2

1 2 1 2 1 2( ; ) l m l ml l m

g g l l m r Y Y r

1/ 2

1 2 1 2 1 2

2 1( ; ) ( ; 0) ( ; )

4l

lg l l m r C l l l mm g l l l r

1 2

1 2

1 2 1 2 1 2( ; ) l m l ml l m

g g l l m r Y Y r

removing the m dependence:

reconstruction

EXAMPLE: dipoles in LJ spheres

r

φθ2

How to present the results of MC simulations?

Conclusions and considerations for the future

- we have briefly reviewed the statistical mechanics of complex liquids

- in order to reduce the number of interaction terms that have to be evaluated in each simulation step a spherical multipole expansion of the electrostatic interaction energy was made

- the basics of the Monte-Carlo method for simulation of ensembles of anisotropic particles were provided along with useful methods for representing the results of such simulations.

- finally results of MC simulations of dipoles embedded in Lennard-Jones spheres were briefly presented.

- employ such simulations to study biophysical processes, such as the hydrophobic effect

- possibility of including polarization effects basis for developing a polarizable water model for biomolecular simulations