344
P. Xiang and Y. A. Wang
Using similar derivations as shown in Sect. 2, we get the variation of the Lieb functional:
𝛿F L [𝜌 0 ] =
∑
k
s k
⟨[ E 0
N
− v(𝐫)
]
𝛿𝜌
k
(𝐫)
⟩
+
∑
k
s k
inf
Ψ
k
0
+í µí»¿Ψ k →𝜌
k
0
+𝛿𝜌 k
⟨𝛿Ψ
k
| ̂
H − E 0 |𝛿Ψ
k
⟩
=
⟨[ E 0
N
− v(𝐫)
]
𝛿𝜌 0 (𝐫)
⟩
+
∑
k
s k
inf
Ψ
k
0
+í µí»¿Ψ k →𝜌
k
0
+𝛿𝜌 k
⟨𝛿Ψ
k
| ̂
H − E 0 |𝛿Ψ
k
⟩ , (66)
based on Eq. (27), where the total density variation 𝛿𝜌 is a linear combination of
individual variation of 𝜌
k
0
,
𝛿𝜌 =
∑
k
s k 𝛿𝜌
k
,
(67)
where s k is the same as that in Eq. (27). Then, the condition for the Lieb functional
to be Fréchet differentiable is
∑
k
s k
inf
Ψ
k
0
+í µí»¿Ψ k →𝜌
k
0
+𝛿𝜌 k
⟨𝛿Ψ
k
| ̂
H − E 0 |𝛿Ψ
k
⟩
||𝛿𝜌||
→ 0 , as ||𝛿𝜌|| → 0 .
(68)
For each PS-v-representable density 𝜌
k
0
in the expansion of the E-v-representable
density 𝜌 0 , we use the same kind of variational path as shown before. Let the density
𝜌
k
p = 𝜌
k
0
+ 𝛿𝜌
k along the variational path be generated by the wavefunction:
Ψ
k
p = Ψ
k
0
+ 𝛽Ψ
k
 ,
(69)
as 𝛽 approaches zero. The corresponding total density variation is
𝛿𝜌 = 𝛽
2
∑
k
s k 𝜌
k

.
(70)
Along each variational path, let the trial wavefunction to be
̃
Ψ
k
= Ψ
k
0
+ 𝜆 k Ψ
k
t .
(71)
By the same analysis, we can show that each scaling parameter 𝜆 k must satisfy
𝜆
2
k
= 𝛽
2
,
(72)
and each trial wavefunction Ψ
k
t can only be expanded in the corresponding  space,
Ψ
k
t
=

∑
i
c
k
i
Ψ
k
i
.
(73)