Empirical Potential Structure Refinement

European Synchrotron Radiation Facility Data Analysis Tutorial23rd - 24th October 2006

Daniel BowronISIS Facility

CLRC Rutherford Appleton LaboratoryChilton, Didcot

OX11 0QXUnited Kingdom

EPSR tutorial

Empirical Potential Structure Refinement• A computational method for building atomic and molecular structural models of disordered materials, such as liquids and glasses, that are consistent with available structural data and known physical/chemical constraints.

Three main ingredients:

(1) Experimental data (2) Monte Carlo computer simulation(3) Known constraints such as density and molecular structures

One principal output:

A 3-dimensional structural modelEPSR tutorial

Monte Carlo computer simulations: What's needed?

(1) A computer representation of box of atoms and molecules – essentially a store of coordinates.

(2) A set of potential energy functions to model the interactions between the atoms and molecules in the box.

(3) A set of rules by which the atoms, molecules, functional groups within the model are moved e.g.

(a) Atomic translations(b) Molecule translations(c) Molecule rotations(d) Molecule functional group rotations(e) Torsional operations on molecules etc.

(4) A set of tools to interrogate the development of the interatomic and intermolecular structures that the model will produce.

EPSR tutorial

A box of atoms and molecules:

L

Cubic box of side length L/ÅA store of atomic coordinates:

Constraints on first and second neighbour distances within specific groups of atoms are one means by which it is possible to define molecules and maintain their shapes.

Key parameters for the box are:

(1) The density of atoms

(2) The total energy, U

(3) The total pressure

EPSR tutorial

Forces between atoms and molecules:

) ) ))) )))

) ) ) )

) )))

The interaction between atoms involves only the interatomic forces and electric charges for the case of ions.

Molecules are more complex. Not only do we have to take account of inter-atomic forces between the atoms or charged groups of the molecule, but also the sum of their forces on other molecules.

Inter-atomic forces

Intra-molecular forces = inter-atomic forces within a molecular unit.

Inter-molecular forces

EPSR tutorial

Forces between atoms and molecules:

ULJ =4

[

r 12

−r

6]

=12

=1/2

The force between atom α (σα,εα) and atom β (σβ,εβ) can be represented by the derivative of the Lennard-Jones potential shown below:

Lorentz-Berthelot mixing rules for calculating σ and ε between different types of atom

EPSR tutorial

EPSR interaction potentials 1: The intramolecular harmonic potential

U intra=C∑i∑≠ rii−d

2

2w2

w2 =

d

1/2 =

M M M M

Average interatomic distance Reduced mass

Typically 114Å-1 amu-1

Broadening function to avoid having to refine an independentDebye-Waller factor for each intramolecular distance

EPSR tutorial

EPSR interaction potentials 1: The intramolecular harmonic potential

0.0 0.5 1.0 1.5 2.0 2.5 3.00

20

40

60

80

100

U intra

(Si-O

)

r(Å)

U intra =C∑i ∑≠ ri i−d

2

2w2

C = 114 Å-1 amu-1

d2αβ = 1.6Å

w2 = 1.6/((28×16)/(28+16))1/2

EPSR tutorial

EPSR interaction potentials 2: The intermolecular reference potential

U ref=12∑i , j≠ i ∑ ,4[

ri j

n

−

ri j

6

]q q

40 ri j

Lennard-Jones potential + Coulomb charges

Typical values for SPC/E water are:

εOO= 0.65 kJmol-1 εOH= 0 εHH= 0

n = 12 σOO= 3.166Å qO= -0.8476e qH= +0.4238e

EPSR tutorial

EPSR interaction potentials 2: The intermolecular reference potential

U ref=12∑i , j≠ i ∑ ,4[

ri j

n

−

ri j

6

]q q

40 ri j

2 4 6 8 10-0.5

0.0

0.5

1.0

1.5

2.0

U L-J i

nter (S

PC/E

Owa

ter-O

water

)

r(Å)0 2 4 6 8 10

0

2

4

6

8

10

U Coul

omb

Inter

(lik

e cha

rges

)

r(Å)EPSR tutorial

Computer simulation in a cubic box with periodic boundary conditions

If a molecule moves outof the simulation box onone side

It moves back into the boxon the opposite side

EPSR tutorial

Computer simulation in a cubic box with periodic boundary conditionsMinimum image convention

The forces acting on a particle are calculated as though it is in the centre of a box

Minimum image with no radial cutoff

Radial cutoffof potentials

EPSR tutorial

EPSR interaction potentials 2: The intermolecular truncation function

T r =0.51−sin2 r− rminpt

rmaxpt− rminpt r2 rminpt− rmaxpt T r =0T r =1.0

2 rminpt− rmaxpt r rmaxpt

r rmaxpt

To save computation time it is necessary to truncate the intermolecular potential, or more precisely the derivative of the potential at a reasonable radius value from the atomic site. This is done using a smoothly decaying function of the form:

Rminpt

= 9

Rmaxpt

= 12

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

T(r)

r (Å)EPSR tutorial

Monte Carlo computer simulations: How to move an atom or molecule.

In the Monte Carlo method, the process of moving an atom or a molecule is dependent on the determination of a series of random numbers.

An example: we wish to move atom A from a position (x,y,z) to a position (X,Y,Z), where the new position is determined at random.

X = x + (RANDOM NUMBER BETWEEN -1 and 1)*STEP SIZE

Y= y + (RANDOM NUMBER BETWEEN -1 and 1)*STEP SIZE

Z= z + (RANDOM NUMBER BETWEEN -1 and 1)*STEP SIZE

Where "STEP SIZE" is usually fixed at the start of the modelling process.

(x,y,z)

(X,Y,Z)

BUT! Do we accept this move? EPSR tutorial

Monte Carlo computer simulations: move acceptance

The decision whether a particular atom or molecule move in the modelling process is accepted or not, is dependent on how it changes the total potential energy of the model box of atoms.

Case 1: The total potential energy U, decreases ⇒ move accepted

Case 2: The total potential energy U, increases ⇒ move rejected .... unless

exp−UkB T RANDOM NUMBER 0−1

EPSR tutorial

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Si-Si 3.14Å

O-O 2.65Å

Si-O 1.66Å

SiO

2 g(r)

r (Å)

2θλ

Sample DetectorQ=

4

sin

Experimental data: The diffraction experiment

0 10 20 30 40 50

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

SiO 2 S

(Q)

S(Q) (Å-1)

F.T

EPSR tutorial

D Q =∑ ,≥ 2− c c b b SQ

SQ −1=4∫0∞ r2 [gr −1]

sin Qr Qr

dr

Experimental data: The diffraction experiment

The total structure factor Atomic concentrations and scattering lengths

Partial structure factors

Atomic density Partial pair distribution functions

EPSR tutorial

5 10 15 20-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

S HH(Q

)-1

Q(Å-1)5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

1.5S O

H(Q

)-1

Q(Å-1)5 10 15 20

-3

-2

-1

0

1

2

S OO(Q

)-1

Q(Å-1)

Partial structure factors: The case for liquid water

H-H O-H O-O

EPSR tutorial

0 2 4 60

2

4

6

12

14

16 Water 298K

3.84Å

4.50Å

3.29Å

2.36Å1.53Å

2.73Å

1.78Å

0.98Å

OO

OH

HH

g(r)

r(Å)0 2 4 6

0

2

4

6

12

14

16 Water 298K

3.84Å

4.50Å

3.29Å

2.36Å1.53Å

2.73Å

1.78Å

0.98Å

OO

OH

HHg(

r)

r(Å)

2 atoms4.3 atoms

1.9 atoms 4.4 atoms

N

r1r2

=4 c∫r1

r2r2 g r dr

Structural information: liquid water

Access to interatomic:DISTANCESANGLESCOORDINATION NUMBERS

EPSR tutorial

From experiment to computer simulation: The potential of mean force

0 2 4 60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

g HH(r)

r (Å)0 2 4 6

0

2

4

6

8

10

12

14

g OH(r)

r (Å)0 2 4 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

g OO(r)

r (Å)

0 1 2 3 4 5 6-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

� HH(r)

=-ln

[gHH

(r)]

r (Å)0 1 2 3 4 5 6

-1

0

1

2

3

� OH(r)

=-ln

[gOH

(r)]

r (Å)0 1 2 3 4 5 6

-1

0

1

2

3

4

5

� OO(r)

=-ln

[gOO

(r)]

r (Å)

r =−kB T ln g r

HH OH OO

r =−kB T ln g r

EPSR tutorial

Potentials of mean force : the core of the EPSR procedure

r =−kB T ln g r

• Purely pairwise potentials

• In general cannot be used in computer simulation because they already contain information arising from many-body cooperative effects i.e. the packing of molecules in the material.

•However, if a reasonable "first guess" potential is known for a system, the potential of mean force can be used to indicate where this model potential needs to be modified to improve agreement with a measured set of site-site pair correlation functions.

EPSR tutorial

Representing the Empirical Potential – 1

Expand the empirical potential in a series of Poisson functions

U EP r =kB T ∑kCk pn k

r , r

The Ck are real and can be positive or negative and σ

r is a width function. The

function pn(r,σ) has an exact 3-dimensional Fourier transform to Q-space:

pn kr , r=

14 r

3 n2! r r

n

exp[−r r]

pn Q , =4∫ pn r , exp iQ⋅r dr

EPSR tutorial

pn Q , =4∫ pn r , exp iQ⋅r dr

pn Q , = 1

n2 1Q2 2n 4 [2cos n

1−Q2 2Q

sin n]=arctan Q

Representing the Empirical Potential – 2

The Ck can thus be estimated from the diffraction data by fitting a series of the

form:

U EP Q =∑kCk pn k

Q , Q

EPSR tutorial

Linking the Empirical Potential to the experimental dataThe algorithm to run EPSR can now be summarized as:

(1) Establish a reasonable reference potential U0αβ(r)

(2) Run a Monte Carlo simulation until equilibrium is reached(3) Calculate a model SS

αβ(Q)(4) Compare this with the experimental data SD

αβ(Q) and derive a difference function

(5) Continue to run the simulation using this new potential to equilibrium(6) Repeat steps (3), (4) and (5) until:

SSαβ(Q) = SD

αβ(Q)

(7) Continue to run the simulation and accumulate ensemble information

Q =∑kCk pn k

Q , Q

EPSR tutorial

Fi Q =∑ j=1, Nwij S j Q

w' ij= f wij

w' ij=1− f i− M , j

1≤ i≤M

M i≤M N

Pii '=∑ j=1, Nw' ij w ji '

−1−ii '

The fit to an experimental data set can be written as a weighted sum over the pairs of atoms of the relevant simulated structure factors

The task is to invert the matrix wij and refine the empirical potential that is

driving the simulation of the experimental dataIntroduce a confidence factor f (0 ≤ f < 1) and form a modified matrix w'

ij

where we accept input from experimental data with confidence f and input from the simulation with confidence ( 1-f )

Now seek the inverse matrix w-1ji noting that P

ii' is unitary

i = data setj = αβ pair

EPSR tutorial

U m j r =kB T ∑k

Ck , m j pn k

r , r

At the mth iteration, the empirical potential for any particular atom pair j is:

At the mth iteration, the difference between the data and the fit is calculated:

Di Q −Fi Q

At the beginning, when m=1, the Ck , m j =0

It is then possible to calculate the difference coefficients: Ck i

These coefficients are then accumulated in the potential coefficients:

Ck , m1 j =Ck , m

j ∑i=1, Mw ji

−1Ck i

And these new coefficients are then fed back as a modified empirical potential

EPSR tutorial

The cycle is repeated a large number of times until

Ck i becomes very small and the EP ceases to change 1.

2. U =4∑ j=1, N∣U m

j r ∣g j r r2 dr

Reaches a predefined limit that usually signifies the presence of systematic errors in the data

Once this criterion is reached the partial structure factors from the simulation can then be calculated as:

T j Q =∑i=1, Mw ji

−1 f Di Q ∑i= M 1, M Nw ji

−1 1− f S i− M Q

And other ensemble average functions accumulated

OR

EPSR tutorial

U N =U

O kT ln g r g

D r

U O r ≈U

N r

g r ≈gD r

0 5 10 15-12-10-8-6-4-202

0 5 10 15-202468

10

0 5 10 15-4-202468

101214

TBA

Met

hyl s

ubs

HH S

(Q)

Q(Å-1)

TBA

Met

hyl s

ubs

XH S

(Q)

Q(Å-1)

TBA

Met

hyl s

ubs

XX S

(Q)

Q(Å-1)

Constraints such as:Density

Molecular geometry

Experimental data

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

g CC-C

C(r)

r(Å)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

g CC-C(

r)

r(Å)

g CC-M(r)

r(Å)

g CC-O(r)

r(Å)

g CC-H(

r)

r(Å)

g CC-O

W(r)

r(Å)

g CC-H

W(r)

r(Å)

g C-C(r

)

r(Å)

g C-M(

r)

r(Å)

g C-O(r

)

r(Å)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

g C-H(r

)

r(Å)

g C-OW

(r)

r(Å)

g C-HW

(r)

r(Å)

g M-M(r

)

r(Å)

g M-O(r

)

r(Å)

g M-H(r

)

r(Å)

g M-OW

(r)

r(Å)

g M-HW

(r)

r(Å)

g O-O(r

)

r(Å)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 100

1

2

3

4

5

6

7

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 2 4 6 8 100

1

2

3

4

5

0 2 4 6 8 100

1

2

3

4

5

0 2 4 6 8 100

2

4

6

8

10

12

14

16

0 2 4 6 8 100

1

2

3

4

5

6

0 2 4 6 8 100

1

2

3

4

5

6

g O-H(r

)

r(Å)

g O-OW

(r)

r(Å)

g O-HW

(r)

r(Å)

g H-H(r

)

r(Å)

g H-OW

(r)

r(Å)

g H-HW

(r)

r(Å)

g OW-O

W(r)

r(Å)

g OW-H

W(r)

r(Å)

g HW-H

W(r)

r(Å)

INPUT OUTPUTA wealth of structural

information

EPSR

EPSR: a summary

EPSR tutorial

Useful referencesComputer Simulation of Liquids, M.P.Allen and D.J.Tildesley, (1987) Oxford University Press, Oxford and New York

Empirical Potential Monte Carlo simulation of fluid structureA.K.Soper, Chemical Physics, 202, (1996) 295-306

The radial distribution functions of water and ice from 220 to 673K and at pressures up to 400MPaA.K.Soper, Chemical Physics, 258, (2000) 121-137

Tests of the empirical potential structure refinement method and a new method of application to neutron diffraction data on waterA.K.Soper, Molecular Physics, 99 (2001) 1503-1516

Partial structure factors from disordered materials diffraction data: An approach using empirical potential structure refinementA.K.Soper, Physical Review B, 72, (2005) 104204

EPSR tutorial