1

CE 530 Molecular Simulation

Lecture 15

Long-range forces and Ewald sum

David A. Kofke

Department of Chemical Engineering

SUNY Buffalo

kofke@eng.buffalo.edu

2

Review

Intermolecular forces arise from quantum mechanics• too complex to include in lengthy simulations of bulk phases

Empirical forms give simple formulas to approximate behavior• intramolecular forms: bend, stretch, torsion

• intermolecular: van der Waals, electrostatics, polarization

Unlike-atom interactions weak link in quantitative work

3

Truncating the Potential Bulk system modeled via

periodic boundary condition• not feasible to include

interactions with all images

• must truncate potential at half the box length (at most) to have all separations treated consistently

Contributions from distant separations may be important

These two are same distance from central atom, yet:

Black atom interactsGreen atom does not

These two are nearest images for central atom

Only interactions considered

4

Truncating the Potential

Potential truncation introduces discontinuity• Corresponds to an infinite force

• Problematic for MD simulationsruins energy conservation

Shifted potentials• Removes infinite force

• Still discontinuity in force

Shifted-force potentials• Routinely used in MD

For quantitative work need to re-introduce long-range interactions

( ) ( )( )

0c c

sc

u r u r r ru r

r r

( ) ( ) ( )( )

0

c c csf

c

duu r u r r r r r

u r drr r

5

Truncating the Potential

( ) ( )( )

0c c

sc

u r u r r ru r

r r

( ) ( ) ( )( )

0

c c csf

c

duu r u r r r r r

u r drr r

Lennard-Jones example• rc = 2.5

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

2.42.01.61.2Separation, r/

-0.20

-0.15

-0.10

-0.05

0.00

2.52.42.32.22.12.0Separation, r/

u(r) f(r) u(r) u-Shift u(r) f-Shift f(r) f-Shift

6

Radial Distribution Function

Radial distribution function, g(r)• key quantity in statistical mechanics

• quantifies correlation between atom pairs

Definition

Here’s an applet that computes g(r)

( )( )

( )id

r dg r

r d

r

r

Number of atoms at r for ideal gas

Number of atoms at r in actual system

( )id Nr d d

V r r

dr

4

3

2

1

0

543210

Hard-sphere g(r) Low density High density

7

Radial Distribution Function. API

U ser 's P ers pe ctiv e o n the M o lec u la r S im u la tion A P I

S pa ce

In te gra to r

C on tro lle r

M e te rR D F

M e te rFu n ction

M e te rA b s tra ct B ou nda ry C on fig u ra tion

P ha se S p ec ies P o te n tia l D isp lay D ev ice

S im u la tion

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Radial Distribution Function. Java Code

/** * Computes RDF for the current configuration */public double[] currentValue() { iterator.reset(); //prepare iterator of atom pairs for(int i=0; i<nPoints; i++) {y[i] = 0.0;} //zero histogram while(iterator.hasNext()) { //loop over all pairs in phase double r = Math.sqrt(iterator.next().r2()); //get pair separation if(r < xMax) { int index = (int)(r/delr); //determine histogram index y[index]+=2; //add once for each atom } } int n = phase.atomCount(); //compute normalization: divide by double norm = n*n/phase.volume(); //n, and density*(volume of shell) for(int i=0; i<nPoints; i++) {y[i] /= (norm*vShell[i]);} return y;}

public class MeterRDF extends MeterFunction

9

Simple Long-Range Correction

Approximate distant interactions by assuming uniform distribution beyond cutoff: g(r) = 1 r > rcut

Corrections to thermodynamic properties• Internal energy

• Virial

• Chemical potential

2( )42

cut

lrcr

NU u r r dr

Expression for Lennard-Jones model

2 214

6cut

lrcr

duP r r dr

dr

2( )4 2

cut

lrclrc

r

Uu r r dr

N

9 338

39

LJlrc

c cU N

r r

9 32 332 3

9 2LJ

lrcc c

Pr r

For rc/ = 2.5, these are about 5-10% of the total values

10

Coulombic Long-Range Correction Coulombic interactions must be treated specially

• very long range

• 1/r form does not die off as quickly as volume grows

• finite only because + and – contributions cancel

Methods• Full lattice sum

Here is an applet demonstrating direct approach

Ewald sum

• Treat surroundings as dielectric continuum

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

214

cr

r drr

11

Aside: Fourier Series

Consider periodic function on -L/2, +L/2

A Fourier series provides an equivalent representation of the function

The coefficients are

f(x)

x/ 2L / 2L

One period

102

1

( ) cos sinn nn

f x a a nx b nx

/ 21

/ 2

1

( )cos(2 / )

( )sin(2 / )

L

n LL

L

n LL

a f x nx L dx

b f x nx L dx

12

Fourier Series Example

f(x) is a square wave

/ 21

/ 2

0 / 21 1

/ 2 0

/ 21

/ 2

4

( )cos(2 / )

cos(2 / ) cos(2 / ) 0

( )sin(2 / )

n odd

0 n even

L

n LL

L

L LL

L

n LL

n

a f x nx L dx

nx L dx nx L dx

b f x nx L dx

f(x)

x

13

Fourier Series Example

f(x) is a square wave

/ 21

/ 2

0 / 21 1

/ 2 0

/ 21

/ 2

4

( )cos(2 / )

cos(2 / ) cos(2 / ) 0

( )sin(2 / )

n odd

0 n even

L

n LL

L

L LL

L

n LL

n

a f x nx L dx

nx L dx nx L dx

b f x nx L dx

f(x)

x

n = 1

14

Fourier Series Example

f(x) is a square wave

/ 21

/ 2

0 / 21 1

/ 2 0

/ 21

/ 2

4

( )cos(2 / )

cos(2 / ) cos(2 / ) 0

( )sin(2 / )

n odd

0 n even

L

n LL

L

L LL

L

n LL

n

a f x nx L dx

nx L dx nx L dx

b f x nx L dx

f(x)

x

n = 1, 3

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Fourier Series Example

f(x) is a square wave

/ 21

/ 2

0 / 21 1

/ 2 0

/ 21

/ 2

4

( )cos(2 / )

cos(2 / ) cos(2 / ) 0

( )sin(2 / )

n odd

0 n even

L

n LL

L

L LL

L

n LL

n

a f x nx L dx

nx L dx nx L dx

b f x nx L dx

f(x)

x

n = 1, 3, 5

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Fourier Series Example

f(x) is a square wave

/ 21

/ 2

0 / 21 1

/ 2 0

/ 21

/ 2

4

( )cos(2 / )

cos(2 / ) cos(2 / ) 0

( )sin(2 / )

n odd

0 n even

L

n LL

L

L LL

L

n LL

n

a f x nx L dx

nx L dx nx L dx

b f x nx L dx

f(x)

x

n = 1, 3, 5, 7

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Fourier Representation

The set of Fourier-space coefficients bn contain complete information about the function

Although f(x) is periodic to infinity, bn is non-negligible over only a finite range

Sometimes the Fourier representation is more convenient to use1.2

1.0

0.8

0.6

0.4

0.2

bn

403020100

n

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Convergence of Fourier Sum

If f(x) = sin(2kx/L), transform is simple• bn = 1 for n = k

• bn = 0 otherwise

f(x)

x

1.0

0.8

0.6

0.4

0.2

0.0

bn

403020100

n

Converges very quickly!

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Observations on Fourier Sum

Smooth functions f(x) require few coefficients bn

Sharp functions (square wave) require more coefficients Large-n coefficients describe high-frequency behavior of f(x)

• large n = short wavelength

Small-n coefficients describe low-frequency behavior• small n = long wavelength

• e.g., n = 0 coefficient is simple average of f(x)

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Fourier Transform As L increases, f(x) becomes less periodic Fourier transform arrives in limit of L Compact form obtained with exponential form of cos/sin

Useful relations• derivative

• convolution

2 2102

1

( ) cos sinnx nxn nL L

n

f x a a b

/ 21

/ 2

1

( )cos(2 / )

( )sin(2 / )

L

nL

L

nL

a f x nx L dx

b f x nx L dx

2ˆ( ) ( ) ikxf x f k e dk

2ˆ ( ) ( ) ikxf k f x e dx

inverse

forward

cos sinie i

an = real part of transform

bn = imaginary part

( ) ˆ( ) ( ) ( 2 ) ( )m mf x k ik f k ˆ ˆ( ) ( ) ( ) ( ) ( )f t g x t dt k f k g k

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Fourier Transform Example Gaussian

Transform is also a Gaussian!

Width of transform is reciprocal of width of function• k-space is “reciprocal” space

• sharp f(x) requires more values of F(k) for good representation

• (x-xo) transforms into a sine/cosine wave of frequency xo:

1/ 2 22 1ˆ( ) exp2

kf k

2

( ) exp2 2

xf x

f(x)

x

F(k)

x

2ˆ( ) oikxk e

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Fourier Transform Relevance

Many features of statistical-mechanical systems are described in k-space• structure

• transport behavior

• electrostatics

This description focuses on the correlations shown over a particular length scale (depending on k)

Macroscopic observables are recovered in the k 0 limit Corresponding treatment is applied in the time/frequency

domains

23

Review of Basic Electrostatics

Force between charges In terms of electric field Static electric field satisfies

Charge density (r)• for point charge q2:

Electrostatic potential• zero curl implies E can be written

• potential energy of charge q1 at r, relative to position at infinity

Poisson’s equation•

1 22

ˆq q

rF r

1( ) ( )qF r E r

( ) 4 ( )

( ) 0

E r r

E r

2( ) ( )q r r

( ) ( ) E r r

2 4

1( ) ( )u q r r

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Ewald Sum

We want to sum the interaction energy of each charge in the central volume with all images of the other charges• express in terms of electostatic potential

• the charge density creating the potential is

• this is a periodic function (of period L), but it is very sharpFourier representation would never converge

+ ++

--

-+ ++

--

- + ++

--

-+ ++

--

- + ++

--

-

+ ++

--

-+ ++

--

- + ++

--

-+ ++

--

- + ++

--

-

+ ++

--

-+ ++

--

- + ++

--

-+ ++

--

- + ++

--

-

+ ++

--

-+ ++

--

- + ++

--

-+ ++

--

- + ++

--

-

+ ++

--

-+ ++

--

- + ++

--

-+ ++

--

- + ++

--

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charge iin centralvolume

( )q i iU q r

,image in

vectors

( ) ( )

( )

jj

j jj

q

q L

n n

n

r r

r r n

25

Ewald Sum: Fourier 1.

Compute field instead by smearing all the charges

Electrostatic potential via Poisson equation• direct space form

• reciprocal space

Fourier transform the charge density

23/ 2( ) ( / ) exp ( )j jj

q L n

r r r n

include n = 0

2 ( ) 4 ( ) r r2 ( ) 4 ( )k k k

/ 21

/ 2

1

( )cos(2 / )

( )sin(2 / )

L

n LL

L

n LL

a f x nx L dx

b f x nx L dx

2

1

/ 41

( ) ( )

j

iV

V

i kjV

j

d e

q e e

k r

k r

k r r

Large takes back to function

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Ewald Sum. Fourier 2.

Use Poisson’s equation for electrostatic potential

Invert transform to recover real-space potential

• in principle requires sum over infinite number of wave vectors k

• but reciprocal Gaussian goes to zero quickly if is small

102

1

( ) cos sinn n

n

f x a a nx b nx

2

4( ) ( )

k

k k

2

0

( ) / 42

0

( ) ( )

41 j

i

k

ij k

k j

e

qe e

V k

k r

k r r

r k

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Ewald Sum. Fourier 3.

The electrostatic energy can now be obtained• for point charges in potential of smeared charges

Two corrections are needed• self interaction

• correct for smearing

2

2

12

( )/ 412 2 2

0 ,

2/ 412 2

0

( )

4

4( )

i j

q i ii

ii jk

i j

k

U q

q qVe e

k V

Ve

k

k r r

k

k

r

k

product of identical sums

1( ) ji

jj

q eV

k rk

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Ewald Sum. Self Interaction 1. In Ewald sum, each point charge is replaced by smeared Gaussian centered on

that charge• this is done to estimate the electrostatic potential field

All point charges interact with the resulting field to yield the potential energy• This means that the point charge

interacts with its smeared representation

• We need to subtract this

x

x

x

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Ewald Sum. Self Interaction 2. We work in real space to deal with the self term

• Poisson’s equation for the electrostatic potential due to a single smeared charge

• The solution is

• In particular, at r = 0

• The self-correction subtracts this for each charge

2 ( ) 4 ( ) r r23/ 2( ) ( / ) expj jq

r r r

( ) jqr erf r

r

1/ 2(0) 2 ( / )jq

1

2

12

2

(0)self jj

jj

U q

q

independent of configuration

2 22

1r

r rr

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Ewald Sum. Smearing Correction 1.

We add the correct field and subtract the approximate one to correct for the smearing

This field is short ranged for large (narrow Gaussians)• can view as point charges surrounded by shielding

countercharge distribution

( ) ( ) ( )p Gj jj

j jj

j j

jj

j

q qerf

qerfc

r r r

r rr r r r

r rr r

x

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Ewald Sum. Smearing Correction 2.

Sum interaction of all charges with field correction• convenient to stay in real space

• usually is chosen so that sum converges within central image

Total Coulomb energy

• each term depends on , but the sum is independent of itif enough lattice vectors are used in the reciprocal- and real-space sums

Here is an applet that demonstrate the Ewald method

12

12

( )i j iji j

i jij

iji j

U q r

q qerfc r

r

n

n

( ) ) )c q selfU U U U

32

Ewald Method. Comments

Basic form requires an O(N2) calculation• efficiency can be introduced to reduce to O(N3/2)

• good value of is 5L, but should check for given application

• can be extended to sum point dipoles

Other methods are in common use• reaction field

• particle-particle/particle mesh

• fast multipole

33

Summary

Contributions from distant interactions cannot be neglected• potential truncated at no more than half box length

• treat long-range assuming uniform radial distribution function

Coulombic interactions require explicit summing of images• too costly to perform direct sum

• Ewald method is more efficientsmear charges to approximate electrostatic field

simple correction for self interaction

real-space correction for smearing