Analytical Expression for Hessian

Noting that ψ in = ψ ni , r in = r ni and r in
α
as
=
⎡ ⎛
⎢
⎣ζ ⎝
ψ ml
⎞
⎠
m=i
l=m
ζ−1 ∂ψ mi
∂r mi
rmi α
rmi ⎤
⎥
= −rin α ,we can rewrite expression 4
⎡ ⎛
⎢
⎦ + ⎣ζ ⎝
ψ il
⎞ζ−1
⎛
⎠ ⎝
l=i
n=i
∂ψ ni
∂r ni
r ni
α
⎞⎤
rni ⎠⎥
⎦. (5)
The summation and the product can be interchanged in the second term of
expression 5 to obtain
=
⎡ ⎛
⎢
⎣ζ ⎝
ψ ml
⎞ζ−1
∂ψ mi
⎠
∂rmi rmi α
rmi ⎤
⎥
⎦ +
⎡ ⎛
⎢
⎣ζ ⎝
ψ il
⎞ζ−1
∂ψ ni
⎠
∂rni m=i
l=m
n=i
Expression 6 can be re-written as a single sum over m
⎡⎛
∂U EAM
∂x i α
∂U EAM
∂x i α
where χ mi
1 ≡
= ζ
⎢⎜
⎢⎜
⎢⎜
⎢⎜
⎢⎜
m=i⎣⎝
= ζ
χ mi
m=i
⎛
⎝
ψ ml
⎞
⎠
ζ−1
l=m
sum over neighbours of m
1 χmi 2
⎛⎛
⎜
⎝⎝
ψ ml
⎞
⎠
l=m
+
l=i
⎛
⎝
ψ il
⎞
⎠
ζ−1
l=i
sum over neighbours of i
⎞
⎟ mi
⎟ ∂ψ
⎟ ∂r
⎠
mi
rni α
rni ⎤
⎥
⎦. (6)
r mi
α
r mi
⎤
⎥
⎥,
⎥
⎦
, (7)
ζ−1
⎛
+ ⎝
ψ il
⎞ζ−1
⎠
⎞
l=i
⎟
⎠ and χ mi
2
∂ψmi
≡ ∂rmi r mi
α
rmi . Note
that (unlike χ mi
2 ) χ mi
1 is not a function of r mi alone but depends on the configuration
of the system. It can be checked that expression 7 reduces to the
expression C3 of [1] for the case of binary potential where ζ = 1.
The expression for the Hessian (due to the multi-body part alone) is
EAM ∂U
=
,
−H ij
αβ ≡ ∂2 U EAM
∂x j
β ∂xi α
∂
∂x j
β
= ζ ∂
∂x m=i
j
= ζ
m=i
∂xi α
mi
χ1 χ
β
mi
2 ,
χ mi∂χ
1
mi
2
∂x j
+ ζ
β
χ
m=i
mi∂χ
2
mi
1
∂x j
,
β
= A ij
αβ + Bij
αβ . (8)
2