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Appendix A
The Derivatives of the DSM Energy
The first and second derivatives of the DSM energy model with respect to c is found recalling that
and
DSM
( ) ( )
( ) ( ) 2Tr
E c = E D + 2TrFD δ , (A-1)
E D = E D0 + DF + 0 + TrDF, + +
(A-2)
n
D = c ( D −D ), (A-3)
+
∑
i=
1
The two terms in Eq. (A-1) is evaluated one by one:
and
∂E
∂c
( D )
x
i
i
0
D δ = 3DSD −2DSDSD −D. (A-4)
= Tr DF − Tr DF + Tr DF + Tr DF−Tr DF −Tr
DF (A-5)
x 0 0 x x x
0 0
∂
∂F
∂D
2TrFDδ
= 2Tr Dδ
+ 2TrF
∂c ∂c ∂c
x x x
∂Dδ
= 2TrFD
x δ + 2Tr F ,
∂c
x
δ
(A-6)
where
∂D
∂
δ
c x
= 3DSD + 3D SD −2DSDSD −2DSD SD −2D SDSD −D . (A-7)
The second derivative is found in the same manner
∂
where
2
E
∂c
x
( D )
∂c
y
x x x x x x
= 2TrDF + TrDF + TrDF −TrDF −TrDF −TrDF −TrDF, (A-8)
0 0 x y y x 0 x x 0 y 0 0 y
2
2
∂
∂ δ ∂ δ ∂ δ
2Tr δ = 2Tr D x + 2Tr D y + 2Tr
D
x y y x x y
FD F F F , (A-9)
∂c ∂c ∂c ∂c ∂c ∂c
2
∂ D
∂c
∂c
x
δ
y
= 3D SD + 3D SD −2DSD SD −2D SDSD −2DSD SD
y x x y y x y x x y
−2DSDSD−2DSDSD −2 DSDSD.
y x x y x y
(A-10)
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