Analytical Expression for Hessian

Expression for A ij
αβ :
A ij
αβ in expression 8 can be further simplified using
∂χ mi
2
∂x j
β
= rmi α
rmi ∂
∂x j
mi ∂ψ
∂r
β
mi
+ ∂ψmi
∂rmi ∂
∂x j
mi rα r
β
mi
,
mi ψrr =
rmi α
+ ψmi
rmi r rmi
∂
α
∂rmi
1
rmi mi ∂r
∂x j +
β
ψmi r
rmi ∂rmi α
∂x j ,
β
ψ
=
mi
rr rmi α ψmi r
−
rmi rmi α
(rmi ) 2
∂rmi ∂x j +
β
ψmi r
rmi ∂
∂x j
i
xα − x
β
m α ,
ψ
=
mi
ji jm
δ − δ + ψmi r
rmi ji
δ δαβ − δ jm
δαβ ,
where ψr ≡ ∂ψ
∂r
final form as
A ij
αβ
= ζ
m=i
χ mi
1
rr rmi α ψmi r
−
rmi rmi α
(rmi ) 2
mi rβ rmi etc. Thus Aij
αβ in expression 8 can be re-written to obtain the
ψmi rr
(rmi ψmi r
2 −
) (rmi ) 3
r mi
α r mi ji jm
β δ − δ
+ ζδαβ χ
m=i
mi
1
A ij
αβ reduces to the expression C4 of [1] for ζ = 1. Further, we can obtain the
expression for the off-diagonal term from equation 10, by setting i = j as
A ij
ψ
αβ = −ζχji 1
ji
r ji
α rji
β − ζδαβχ ji
1
rr
(r ji )
which can also be written as
A ij
ψ
αβ = −ζχij 1
ij
rr
(r ij )
ψji r
2 −
(r ji ) 3
ψij r
2 −
(r ij ) 3
r ij
α rij
ψ
β + δαβ
ij
r
rij
(9)
ji ψr rji
, (11)
i = j, (12)
which reduces to the expression C5 in [1] for ζ = 1. The expression for i = j
can be obtained from equation 10
ψmi r
2 − r mi
α rmi
β + ζδαβ χ miψ
1
mi
r
, (13)
rmi A ii
αβ = ζ χ
m=i
mi
1
ψ mi
rr
(r mi )
(r mi ) 3
which is the generalisation of expression C6 in [1].
3
m=i
mi ψr rmi ji jm
δ − δ
. (10)