1. DEVELOPMENT OF SEMI-EMPIRICAL ATOMISTIC POTENTIALS MS-MEAM
M. I. Baskes
Los Alamos National Laboratory
and
University of California, San Diego

2. OUTLINE The Challenge The Concepts The Models Bond energy
Many body effects
Transferability
Reference state
Screening
The Models
Pair potentials
Embedded atom method (EAM)
Modified EAM (MEAM)
Multi-state MEAM (MS-MEAM

3. ATOMISTIC MODELS HAVE TWO PURPOSES (I)
Obtain understanding of physical processes
Model system (empirical) potentials
how specific properties affect collective behavior
dependence of yield strength on stacking fault energy
Semi-empirical potentials (fit to experimental data)
plasticity
phase transformations
First principles
diffusion
This is meant to be a short list of the many physical process that can be examined by atomistics

4. ATOMISTIC MODELS HAVE TWO PURPOSES (II)
Obtain quantitative properties of specific materials
First principles
lattice constant
structural stability
elastic moduli
Semi-empirical potentials (fit to experimental data)
thermal expansion
melting point
yield strength
thermodynamics / free energy / phase diagrams
Many more examples exist

5. IN ORDER TO ACHIEVE THE SECOND PURPOSE WE NEED A METHOD THAT ENCOMPASSES
Accuracy
thermodynamic properties must be known accurately to be useful (a few tenths of a percent of the cohesive energy)
Computational speed
analytic or tabular model
scales linearly with the number of atoms
parallel architecture
We only consider here models that have appropriate speed and then try to improve accuracy

6. CONCEPT: BOND ENERGY
Every pair of atoms is connected by a bond (spring)
The bond energy depends on the separation of the atoms
The energy of a material is the sum of the bond energies

7. CONCEPT: MANY BODY EFFECTS
All bonds are not equal
The bond energy also depends on the local environment (coordination)
Coordination / bond length / bond energy are correlated (Pauling)

8. CONCEPT: TRANSFERABILITY
The model will be accurate for all atomic environments
Volume (Nearest neighbor (NN) distance)
Coordination (crystal structure - number of NN)
Defects or strain (loss of symmetry)

9. CONCEPT: REFERENCE STATE (I)
Reference structure
A specific crystal structure
Properties of the reference structure can be obtained from experiment or first principles calculations
Energy vs. volume (NN distance)
Elastic constants
Defect energies
Reference structures have high symmetry
Scaling
energy per atom of the equilibrium reference structure is -1
distance is scaled by the equilibrium NN distance

10. CONCEPT: REFERENCE STATE (II)
Reference path
A specific path connecting 2 reference structures
Properties along the reference path can be obtained from first principles calculations
Energy vs. distance along path
Reference paths encompass low symmetry states
Coordination changes along a reference path
Incorporation of many reference states will facilitate transferability

11. CONCEPT: SCREENING Atomic interactions have a finite range
Radial screening
at a cutoff distance the interactions go to zero (smoothly)
dependant on distance
independent of local geometry
Angular screening
intervening atoms reduce interactions to zero
dependent on local geometry
high compression
Necessary for computational scaling with the number of atoms
Fellows 11/18/2005

12. A PAIR POTENTIAL REPRESENTS ONLY DISTANCE DEPENDENT BONDING
Different Strength
Same Strength

13. MODEL: PAIR POTENTIAL Accuracy Computation Transferable
Volume
Coordination
Defects/strain
Computation
Analytic or tabular
Scales with number of atoms
Parallel architecture
i: all atoms
j: neighbors of atom i
independent of environment
radial screening

14. THE EMBEDDED ATOM METHOD IS SEMI-EMPIRICAL
embedding
energy
host electron
density
pair interaction
UBER
ρ is obtained from a linear superposition of atomic densities
F and ϕ are obtained by fitting to the following properties:
Universal Binding Energy Relationship (UBER)
(lattice constant, bulk modulus, cohesive energy)
Shear moduli
Vacancy formation energy
Structural energy differences (hcp/fcc, bcc/fcc) _

15. MODEL: EAM Accuracy Computation Transferable Analytic or tabular
Volume
Coordination
Defects/strain
Computation
Analytic or tabular
Scales with number of atoms
Parallel architecture
i: all atoms
j: neighbors of atom i
radial screening
depends on environment

16. COMPLEX MATERIALS REQUIRE THE ADDITION OF ANGULAR FORCES
EAM uses a linear superposition of spherically averaged electron densities
MEAM allows the background electron density to depend on the local symmetry     θ 

17. MODEL: MEAM Accuracy Computation Transferable Analytic or tabular
Volume
Coordination
Defects/strain
Computation
Analytic or tabular
Scales with number of atoms
Parallel architecture
Environmental dependence of bonding
Angular screening
Assumed functional forms
embedding function
electron density
background electron density
screening

18. MODIFIED EMBEDDED ATOM METHOD (MEAM)
Universal Binding Energy Relationship
UBER
Background Electron Density
Embedding Function
Pair Potential
12 parameters + angular screening for the pair potential and electron densities

19. CONCEPT OF THE SCREENING ELLIPSE LEADS TO A SIMPLE SCREENING MODEL
screening ellipse defined by C
2y/rik
Cmin and Cmax set limits of screening
2x/rik
goes from 0 to 1 smoothly

20. MULTI-STATE MEAM (MS-MEAM)
Same Functional Form as MEAM
Multiple Reference States
Environmental Dependence of Bonds
Angular Screening
Assumed Functional Forms
Asymptotic embedding function
Background electron density

21. MODEL: MS-MEAM Accuracy Computation Transferable Analytic or tabular
Volume
Coordination
Defects
(we hope!)
Computation
Analytic or tabular
Scales with number of atoms
Parallel architecture
Same functional form as MEAM
Multiple reference states
Environmental dependence of bonds
Angular screening
Assumed functional forms
asymptotic embedding function
background electron density

22. MULTI-STATE MODIFIED EMBEDDED ATOM METHOD (MS-MEAM)
Basic Ansatz
Embedding Function
Background Electron Density
Screening

23. FIRST APPLICATION OF MS-MEAM HAS BEEN COMPLETED
Cu Chosen as Model Material
VASP/GGA-PW Used for First Principles Energy Calculations
~1000 E/V points calculated
M. I. Baskes, S. G. Srinivasan, S. M. Valone, and R. G. Hoagland, Multistate modified embedded atom method, PHYSICAL REVIEW B 75,

24. MS-MEAM EMBEDDING FUNCTION
fcc equilibrium density

25. MS-MEAM ELECTRON DENSITIES
simple smooth functions
negative square densities

26. NEED TO HAVE TWO SETS OF ELECTRON DENSITIES
magnetism
electronic states
charges

27. MS-MEAM SCREENING FUNCTIONS
screening ellipse  k i   j
Fellows 11/18/2005

28. MS-MEAM IS PREDICTIVE FOR ENERGY vs. NN DISTANCE
* used in development of functions
coordination 1-12

29. ENERGY DIFFERENCES FOR EQUAL COORDINATION STRUCTURES ARE SMALL
* used in development of functions
Fellows 11/18/2005

30. ELASTIC CONSTANTS PREDICTED BY MS-MEAM SHOW SHARP INCREASE AT HIGH COMPRESSION
fcc
C44

31. HOMOGENEOUS TRANSFORMATIONS USED TO DETERMINE SCREENING FUNCTIONS
2D-HEX  2D-SQ
FCC  BCC (Bain)
BCC  SC  FCC
(trigonal)

32. TRANSFORMATIONS ARE A SERIOUS TEST OF TRANSFERABILITY
* used in development of functions *

33. CONCLUSIONS
MS-MEAM Has The Potential to be a Fast, Accurate Method of Calculating Atomistic Interactions
Consider MS-MEAM to be a Method for Interpolation/Extrapolation of a FP Data Base
There is No Fitting – Just Direct Calculation From the Data Base
This Method Could Enable Quantitative Thermodynamic Predictions of Multi-component, Multi-phase Materials