Tersoff Potential- to predict structural properties and energetic of a complex system, a new approach to formulation of interatomic potential

1. Tersoff Potential:
Interatomic Potential for
Covalent System
B. Tech. Project
Guide: Prof. Abhijit Chatterjee
Submitted By
Pallavi Kumari (Y9397)
4th Year Undergraduate Student
Department of Chemical Engineering
IIT Kanpur

2. Introduction
To predict structural properties and energetic of a complex system, a new approach to formulation of
interatomic potential is formulated by J.Tersoff. The basic of this approach is that, in real system, the strength
of each bond (i.e., the bond order) depends upon the local environment. In general, atoms with many
neighbors form weaker bonds than an atom with few neighbors. Utilizing this general idea, a new empirical
potential “tersoff potential” is formulated for silicon.
For determination of exact nature of properties and phenomenon like surface reconstructions, diffusion paths
and barriers, reaction co-ordinate and barriers, and thermal and mechanical properties, the total energy of a
system of atoms as a function of the atomic co-ordinate is required.
Among existing models which give an empirical interatomic potential E({r}), the Keating Model is the most
famous. It is roughly analogous to Taylor expansions of the energy about its minimum. The energy of N
interacting particles can be written as
(1)
Where rn is the position of nth atom and Vm is “m-body potential”.
Bond Order
While constructing an accurate potential, it is natural to abandon the use of N-body form. Since the bond
strength or bond order depends on local geometry, an environment dependent bond-order is included in
interatomic potential. The interatomic potential can be written in the form
(2)
Vij = fc (rij) [aij fR (rij) + bij fA (rij)] (3)
where E is total energy of the system. The indices I and j run over the atoms of the system and rij is the
distance from atom I to atom j. The function fR represents a repulsive pair potential which includes the
orthogonalization energy when atomic wave functions overlap, and fA represents an attractive pair potential
associated with bonding. The function fc is a smooth cutoff function to limit the range of potential.
The function bij is a measure of bond order and a monotonically decreasing function of the co-ordination of
atom I and j. The function aij consists of range limiting terms.
The function fR , fA , fc , aij and bij are taken as
fR (r) = A exp (λ1r) (4)
fA (r) = -B exp (λ2r) (5)
The cutoff function fc is a continuous function and has derivative for all r. R is chosen to include only the first-
neighbor shell.

3. fc (r) =
bij =(1+βnξijn)-1/2n (7)
ξij = fc(rik) g(θijk) exp[λ33(rij -rik)3] (8)
g(θ) = 1 + c2/d2 - c2/[d2 +(h-cosθ)2] (9)
Θijk is the bond angle between bonds ij and ik.
aij = (1+αnηijn)-1/2n (10)
ηij = fc(rik) exp[λ33(rij -rik)3] (11)
Parameters
Parameter Value
A 3264.7 eV
B 95.373 eV
λ1 3.2394 A°-1
λ2 1.3258 A°-1
λ3 1.3258 A°-1
α 0
β 0.33675
c 4.8381
d 2.0417
n 22.956
h 0
R 3.0 A°
D 0.2 A°
As α=0, from equation (10) aij = 1
Implementation of Potential Energy and Force
Energy
In equation (1), the factor of ½ takes care that, a bond is considered once. So if we eliminate this factor of ½
by
(12)
Then,
Vij = fc(rij)[aij fR (rij) + fA(rij)] (13)
(6)

4. Since R is chosen to include only the first-neighbor shell, for every i-atom it is wiser to choose only the first-
neighbor shell atom of i-atom as j-atom. The function fR and fA can be calculated easily by retrieving
parameters from input file where all the parameters are stored. For simplification in calculation of function fc
fc = ½ - ½ Sin [ (π/2)*(MAX(rij,Rij-Dij)-Rij)/Dij ] when rij < (Rij+Dij)
For calculation of ξij, again it is wiser to consider only the first-neighbor shell atom of i-atom as k-atom.
Similarly for calculation of ξji, again it is better to consider only the first-neighbor shell atom of j-atom as k-
atom. Cosθ can be easily calculated by dot product of rij and rik.
Force
As the interatomic potential is calculated by taking account of i, j and k-atom, it will contribute to forces on
these atom by
Fiζ = -
Fjζ = -
Fkζ = -
Where ζ = x,y,z and Fiζ is force on i-atom in ζ direction and so on.
Proceeding for Fiζ and putting aij = 1
Fiζ = - (14)
Leaving the summation part as it will be taken care by recursive addition within the loop
Fiζ = - * * - * [ *
* ] (15a)
Fjζ = - * * - * [ *
* ] (15b)

5. Results
The above model is applied to a close packed Silicon cube. The energy calculated from the above model is
found to be -9590.9802 eV. The model has 1000 atoms and each bond is shared between 2 atoms. Hence
Enery of one atom = (-9590.9802)/(2*1000) = -4.7954 eV
As this is an uniform packing, distance between any two adjacent atoms will be the same. Retrieving that
distance from the code which is 2.3512 A°, and assuming Si-Si bond to be tetrahedral & taking their bond
angle to be 109.4°, rjk is calculated.
rjk = = 3.835
cos θ = -0.33
From equation 9; g(θ) = 1.41147
From equation 8; ξij = 3.4244
From equation 7; bij = 0.9305
As neighborhood of every atom is same, bji = bij = 0.9305
From equation 4; fR = 3264.7 exp (-3.2394* 2.3512) = 1.6068 eV
From equation 5; fA = -97.373 exp(-1.3258*2.3512) = -4.3113 eV
From equation 3; Energy of one bond = 1*[1.6068 - 0.9305*4.3113] = -2.4047 eV
As each atom has 4 bonds and each bond is shared between 2 atoms;
Energy of one atom = 2*(-2.404) = -4.8094 eV
From literature
Fig: Cohesive energy vs volumeper atom of silicon in the diamond, simple cubic(sc), β-tin(β), simple
hexagonal (sh), bcc and fcc structures.
109.4
i
j
k

6. This graph was re-ploted with the help of pixel, the minimum energy is found to be -4.63 eV. As we are
dealing with a packing in equilibrium, the minimum energy is the energy we were looking for.
Deviation from expected result = (-4.63+4.7954) = 0.1654 eV
Considering a particular pair of atom, Force on one atom if found to be -2.0771x - 2.0771y - 2.0771z
While force on another atom j is found to be 2.0771x + 2.0771y + 2.0771z.
This force can be verified by displacing the atom by Δr and calculating its energy.
Fiζ =-
Acknowledgement
I would like to thank my B.Tech Project Guide Prof. Abhijit Chatterjee for giving me an opportunity to work in
this new field which has helped me envisioned about the deeper aspect of technologies in chemical
engineering. I would also like to acknowledge the contribution of Ms. Shraddha D. Mule to guide me through
this project.
References
1. J. Tersoff; New Empirical approach for the structure and energy of covalent systems; Physical Review
B, Volume 37, Number 12.
2. DonaldW Brenner,Olga A Shenderova, Judith A Harrison, Steven J Stuart, Boris Ni and Susan B
Sinnott; A second-generation reactive empirical bond order (REBO) potential energy expression for
hydrocarbons; Journal Of Physics: Condensed Matter