Analytical Expression for Hessian

constraint l = i. Thus expression 19 simplifies to
B ii
αβ = ζ (ζ − 1)
⎡
⎢
⎣ ψ mi r
r
mi
α
rmi ⎛
⎝
ψ mp
⎞ζ−2
⎠
m=i
⎡⎛
⎢
+ζ (ζ − 1) ⎣⎝
Symmetry checks:
m=i
ψ im
r
Symmetry of the Hessian implies that
H ij
αβ
≡ ∂
∂x j
β
EAM ∂U
∂x i α
p=m
ψ mi
r
rim α
rim ⎞ ⎛
⎠ ⎝
ψ ip
⎞ζ−2
⎛
⎠ ⎝
= ∂
∂x i α
p=i
∂UEAM
∂x j
β
l=i
r mi
β
r mi
ψ il
r
⎤
⎥
⎦
r il
β
⎞⎤
ril ⎠⎥
⎦. (20)
≡ H ji
βα . (21)
Symmetry of A αβ
ij in expressions 12 and 13 is easy to check. We show here the
symmetry of expressions 18 and 20. For comparison, we reproduce expression
18 below,
B ij
αβ
= ζ (ζ − 1)
m=i,j
⎡
⎢
+ζ (ζ − 1) ⎣
⎡
⎢
⎣
ψ ji
r
⎡⎛
⎢
+ζ (ζ − 1) ⎣⎝
m=i
ψ mi
r
r ji
α
r mi
α
r mi
⎛
⎝
p=m
ψ mp⎠
⎞ζ−2
rji ⎛
⎝
ψ jp
⎞ζ−2
⎛
⎠ ⎝
ψ mi
r
p=j
m=j
ψ mj
r
rmi α
rmi ⎞ ⎛
⎠ ⎝
ψ ip
⎞ζ−2
⎠
p=i
From which we obtain by interchanging i ↔ j and α ↔ β,
B ji
βα =
⎡
⎢
ζ (ζ − 1) ⎣ ψ mj
r
r
mj
β
rmj ⎛ ⎝
ψ mp
⎞ζ−2
⎠
m=j,i
⎡
⎢
+ζ (ζ − 1) ⎣
ψ ij
r
⎡⎛
⎢
+ζ (ζ − 1) ⎣⎝
m=j
p=m
r ij
β
rij ⎛
⎝
ψ ip
⎞ζ−2
⎛
⎠ ⎝
ψ mj
r
p=i
m=i
ψ mj
r
ψ ij
r
ψ mi
r
rmj α
rmj ⎞⎛
⎠⎝
ψ jp
⎞ζ−2
⎠
6
p=j
r mj
β
r mj
β
r mj
⎞⎤
rmj ⎠⎥
⎦
ψ mi
r
r ij
β
r ij
r mi
β
⎤
⎥
⎦
⎤
⎥
⎦, i = j.
r mi
α
r mi
⎞⎤
rmi ⎠⎥
⎦
ψ ji
r
r ji
α
r ji
⎤
⎥
⎦
(22)
⎤
⎥
⎦ , j = i.
(23)