1

Ionic and Molecular Liquids:Long-Range Forces and Rigid-Body Constraints

Charusita Chakravarty

Indian Institute of Technology Delhi

2

Electrostatic Interactions

• Ionic liquids: NaCl, SiO2, NH4Cl• mobile charge carriers which are

atomic or molecular entities • Simple ionic melts: model ions as point

charges with Coulombic interactions.

Short-range repulsions control ionic radii.

• Molecular Liquids• Electronic charge distributions show

significant deviations from spherical symmetry

• Can be modeled by: (a) multipole moment expansions or (b) arrays of partial charges

water

benzene

3

Multipole Expansion for Electrostatic Potential

rquadrupola

1cos3

2

dipolar cos

Coulombic 4

3

2

2

r

r

r

Q

Represent the electronic charge distribution of a molecule by a set of multipoles:

4

Range of Electrostatic Interactions

Type Range Energy (kJ/mol) Comment

Ion-Ion 1/r 250

Ion-dipole 1/r2 15

Dipole-Dipole 1/r3 2 Static dipoles (solid phase)

Dipole-quadrupole 1/r4 Fixed Orientation / Linear

Quadrupole-quadrupole 1/r5 Fixed Orientation / Linear

Long-range interactions: Tail correction will diverge for 1/rn interactions with n greater than or equal to 3; therefore minimum image convention cannot be applied

5

Partial Charge Distributions of Some Typical Molecules

109.47

H

O

H

0.95

72 Å

(+0.52e)

(+0.52e)

2(-

1.04e)

1.0 Å

109.47

OO

H H

2(-0.8472e)

(+0.4238e)

(+0.4238e)

TIP4P Water SPC/E Water

Dipole + Quadrupole

Quadrupole

Molecular Nitrogen

6

Array of Point ChargesCoulombic contribution to the potential energy for an array

of N charges that form a charge-neutral system:

Electrostatic potential

• Particle i interacts with

all other charges and their mirror images but not with itself

• Gaussian units

• Cannot apply minimum image convention because

sum converges very slowly

7

Electrostatic Potential

ChargeDistribution

Poisson Equation

Potential

Energy and

Forces

)(r )(r

)rU(rU

and )(

)(4)(2 rr

Linear differential equation:

ii

ii

r

r

)(:potential ticElectrosta

)(:ondistributi Charge

8

Ewald Summation for Point Charges

Co

Point ChargeDistribution:

Converges slowly

)(rP

)(-

)()(

r

rr

G

GP

Screened charge distribution:

Converges fast in real space

Gaussian compensating charge distribution: can be analytically

evaluated in real space

9

Ewald Summation

• Screening a point charge to convert the long-range Coulombic interaction into a short-range interaction

• Evaluating the real-space contribution due to the screened charges

• The Poisson equation in reciprocal space for the compensating screened charge distribution

• Evaluating the reciprocal space contribution

• Self-interaction correction

10

Screening a point charge

)(4

)(1

:ondistributi

chargeGaussian todue potential ticElectrosta

)exp()/()(

:originat centred chargeGaussian Single

/)( and )()(

:originat centred chargepoint Single

2

2

22/3

rr

r

r

rqr

rqrrqr

SGSG

iSG

iSPiSP

)(4)(2 rr

11

Electrostatic potential due to a Gaussian charge distribution

12

)(erfc)/()(erf)/()/(

:density chargeGaussian ngcompensati aby

screened chargepoint of potential ticElectrosta

range-short rrqrrqrq iii

13

Real Space Contribution

ijij

jSR

jij

rerfcr

qr

rqr

)(

:at charge todue at n interactio ticElectrosta

j

jSRjreal rqU )(2

1

:rat potential ticelectrosta with ther positionsat

located q charges ofarray an ofenergy n Interactio

jj

j

i ij

ijij

jireal rerfc

r

qqU

2

1

The value of must be chosen so that the range of interaction of thescreened charges is small enough that a real space cutoff of rc < L/2can be used and the minimum image convention can be applied

14

Poisson Equation in Reciprocal Space

)(42 rr

)(~4)(~2 kkk

)(~

of FT kfkdX

df )(~

of FT kfkdX

fd n

n

n

Fourier series representation of a function in a cubic box with edge length Land volume V under periodic boundary conditions:

),,(2 where)exp()(~

)/1()( zyxl

lllkrkikfVrf

)exp()()( rkirfrdkfV

Fourier coefficients

Poisson equation in real space

Poisson equation in reciprocal space

Reminder: Fourier transforms of derivatives

15

)()(~ rrS

2

4)(~

kkg

i

iiP rrqr )()(~

i

iiP rkqk )exp()(

i

ii rkqk

)exp(4

2

)()()(

~kkgk PP

Unit Point Charge at origin:

Array of point charges

Array of Gaussian charges

i

iiG rrqr )()( )()( rrrdr P

)()(~)(~)(~

kkrkgk P

Green’sFunction Fourier transform

of smearing Function

FT of Point charge Array

Convolution of Point Charge distribution and smearing function

)(~

)(~

)(~

then )()()( :Reminder 321321 xfxfxfxfxfxf

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Fourier Part of Ewald Sum

N

j njjG Lnrrqr

1

223 exp)(

N

ii krk

kk

1

22

4expexp4

)(

Corresponding Electrostatic potential in reciprocal space

Electrostatic potential in real space can be obtained using:

0

)exp()(1

)(k

rkkV

r

Array of N Gaussian point charges with periodic images:

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Fourier Part of Ewald Sum (contd.)

Electrostatic potential in real space

)4/exp()](exp[4

)exp()(1

)(

2

0k 12

0

krrkik

q

rkkV

r

j

N

j

j

k

0

22

2

2

0k 1,2

4exp)(4

2

1

)4/exp()](exp[4

)2/1(

)()2/1(

k

ji

N

ji

ij

iirec

kkkV

krrkiVk

qq

rqU

Reciprocal Space contribution to potential energy

System is embedded in a medium with an inifinite dielectric constant

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Correction for Self-Interaction

rerfr

qr i

Gauss )(

r

qr i

Gauss

2)0(

N

iiselfiSelf rqU

1

)(2

1

N

iiq

1

221

Must remove potential energy contribution due to a continuous Gaussian cloudof charge qi and a point charge qi

located at the centre of the cloud.

Electrostatic potential due to Gaussian centred at origin

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Coulombic Interaction expressed as an Ewald Sum

Reciprocal space

Self-Interaction

Real Space

0

22

24exp

4

2

1

kCoul kk

kVU

N

iiq

1

22

1

N

ji ij

ijji

r

rerfcqq 2

1

Important: For molecules, the self-interaction correction must be modified because partial charges on the same molecule will not interact

20

Accuracy of Ewald Summation

• Convergence parameters:

– Width of Gaussian in real space, – Real space cut-off, rc

– Cutoff in Fourier or reciprocal space, kc=2/Lnc

/

/

sLn

sr

c

c

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Calculating Ewald Sums for NaCl

Na+

Cl-

Liquid Crystal

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Hands-on Exercise:Calculating the Madelung Constant for NaCl

The electrostatic energy of a structure of 2N ions of charge +/- q is

where is the Madelung constant and rnn is the distance between the nearest neighbours.

ijnn r

qN

r

qNU

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Structure Sodium Chloride (NaCl) 1.747565

Cesium Chloride (CsCl) 1.762675

Zinc blende (cubic ZnS) 1.6381

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Rotational and Vibrational Modes of Water

Symmetric Stretch3657cm-1

http

://chsfp

c5.ch

em

.ncsu

.ed

u/~

fran

zen

/CH

79

5N

/lectu

re/X

IV/X

IV.h

tml

Rotational

Constants (cm-1)

A 40.1

B 20.9

C 13.4

Bend1595cm-1

Asymmetric Strech3756cm-1

http

://w

ww

1.ls

bu.

ac.

uk/w

ater

/vib

rat.

htm

Intermolecular

vibrations (cm-1)

Librations 800

OO stretch 200

OOO bend 60

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Molecules: Multiple Time-scales

• Bonded interactions are much stronger than non-bonded interactions

• Intramolecular vibrations have frequencies that are typically an order magnitude greater than those of intermolecular vibrations

• MD/MC: time step will be dictated by fastest vibrational mode

• Fast, intramolecular vibrational modes do not explore much of configuration space- rapid, essentially harmonic, small amplitude motion about equilibrium geometry

• Require efficient sampling of orientational and intermolecular vibrational motions

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Simulation Methods for Molecules

• Freeze out all or some intramolecular modes:– Serve to define vibrationally averaged molecular

structure and are completely decoupled from intermolecular vibrations, librations or rotations

– Rigid-Body Rotations:• Characterize the mass distribution of the molecule by its

moment of inertia tensor• MC: Use orientational moves• MD: Propagate rigid-body equations of motion• Will not work if there are low-frequency vibrational modes

– Apply Geometrical Constraints• MD: SHAKE• MD: RATTLE

• Multiple Time-Scale Algorithms

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Rotations of Rigid-Bodies

H2

N2

CH4SF6

NH3 lCl5

H2O

Ix Iy Iz

Space-fixed (SF) and Body-fixed (BF)axes (Goldstein, Classical Mechanics)

Moments of Inertia of Molecules:Ia < Ib < Ic

Linear: Spherical polar angles: Non-linear:Euler angles: (Atkins, Physical Chemistry)

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Monte Carlo Orientational Moves for Linear Molecules

Orientation of a linear molecule is specified by a unit vector u . To change it by a small amount:

1. Generate a unit vector v with a random orientation. See algorithm to generate random vector on unit sphere

2. Multiply vector v with a scale factor g, which determines acceptance probability of trial orientational move

3. Create a sum vector: t = u + gv

4. Normalise t to obtain trial orientation

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Euler Angles (

http://stackoverflow.com/questionshttp://mathworld.wolfram.com/EulerAngles.html

Euler’s rotation theorem: Any rotation of a rigid-body may bedescribed by a set of three angles

•Rotation, A: Initial orientation of body-fixed axes (X,Y,Z) to final orientation (X’’,Y’’’, Z’)

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Euler angles ((contd.)• Rotate the X-, Y-, and Z-axes about the Z-axis by

resulting in the X'-, Y'-, and Z-axes.

• Rotate the X'-, Y'-, and Z-axes about the X'-axis by resulting in the X'-, Y''-, and Z'-axes.

• Rotate the X'-, Y''-, and Z'-axes about the Z'-axis by resulting in the X''-, Y'''-, and Z'-axes.

Rotation A = BCD, therefore new coordinates are:

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Monte Carlo Orientational Moves for Non-linear Molecules

• Specify the orientation of a rigid body by a quaternion of unit norm Q that may be thought of as a unit vector in four-dimensional space.

31

Applying Geometrical Constraints

)(2

1),( 2 xUxmxxL

q

L

q

L

t )()( qUqKL

Lagrangian Equations of Motion Kinetic Energy Potential Energy

Cartesian coordinates

x

Uxm

x

L

x

L

t

Geometrical Constraints

00

0)(

qt

dqq

d

q

ii i

Define constraint equations and require that system moves tangential to the constraint plane.

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Introduce a new Lagrangian that contains all the constraints:

q

L

q

L

t

''

qLL'

ii

iiii

GF

qq

Uqm

The equations of motion of the constrained system are:

Constraint force acting along coordinate qi

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Bond Length Constraint for Diatomic Molecule

022

21 drr

)(or )(2 211211

rr Grrr

)(or )(2 212212

rr Grrr

m1 m2 Bond constraint

Constraint forces:• lie along bond direction• are equal in magnitude • opposite in direction• do no work

Verlet algorithm:

)()('

)()()()(2)(

2

22

tGm

tttr

tGm

ttF

m

tttrtrttr

ii

i

ii

ii

iii

New positionwith constraint

Unconstrainedposition

Constraint forceson atom i

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)()(

))()()(()(')(

))()()(()(')(

22

21

212

22

212

11

dttrttr

trtrmtttrttr

trtrmtttrttr

i

i

Diatomic molecule (contd)

Three unknowns

In the case of a diatomic molecule, can obtain quadratic equation in . However, cannot be conveniently generalisedfor larger molecules with more constraints.

35

Multiple Constraints

Verlet algorithm for N atoms in the presence of l constraints:

Constraint force actingon i-th atom due to k-thconstraint

Taylor expansion of the constraints with respect to unconstrained positions.Impose the requirement that after one timestep, the constraints must be satisfied.

positions nedunconstrai

using )( ttki Substitute from Taylorexpansion above

36

Multiple Constraints (contd.)

Solution by Matrix Inversion

Need to find the unknown Lagrange multipliers:

Since the Taylor expansion was truncated at first-orderiterative scheme will be required to obtain convergence.

37

SHAKE Algorithm

• Iterative scheme is applied to each constraint in turn:

RATTLE: Similar approach within a velocity Verlet scheme

38

References

• D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications

• A. R. Leach, Molecular Modelling: Principles and Applications

• D. C. Rapaport, The Art of Molecular Dynamics Simulation (Details of how to implement algorithms for molecular systems)

• M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (SHAKE, RATTLE, Ewald subroutines)