Functional Derivatives and Differentiability in Density-Functional Theory
345
Consequently, we have
∑
k
s k
inf
Ψ
k
0
+í µí»¿Ψ k →𝜌
k
0
+𝛿𝜌 k
⟨í µí»¿Ψ k | ̂
H − E 0 |í µí»¿Ψ k ⟩
||𝛿𝜌||
=
∑
k
s k inf
𝜌 k
t →𝜌
k

𝜆
2
k
⟨Ψ k
t
| ̂
H − E 0 |Ψ k ⟩
𝛽 2 ||
∑
k s k 𝛿𝜌
k

||
=
∑
k
s k inf
𝜌 k
t →𝜌
k

⟨Ψ k
t
| ̂
H − E 0 |Ψ k
t
⟩
||
∑
k s k 𝛿𝜌
k

||
,
(74)
which is positive and cannot be zero for any positive semi-definite set of {s k } because
inf
𝜌 k
t →𝜌
k

⟨Ψ
k
t | ̂
H − E 0 |Ψ
k
t ⟩
||
∑
k s k 𝛿𝜌
k

||
> 0 .
(75)
Therefore, the condition for Fréchet differentiability of the Lieb functional is not
fulfilled, and the Lieb functional is not Fréchet differentiable at any E-v-representable
densities.
4 Unconventional Density Variations
In this section, we will examine the implications of our above results.
Density variation in Hilbert space is defined as the difference of a trial density
𝜌(𝐫) from a GS density 𝜌 0 (𝐫),
𝛿𝜌(𝐫) = 𝜌(𝐫) − 𝜌 0 (𝐫) = 𝛽𝜂(𝐫) ,
(76)
where the scaling parameter 𝛽 can take any real value as long as the resultant density
𝜌(𝐫) is not negative,
𝜌(𝐫) = 𝜌 0 (𝐫) + 𝛽𝜂(𝐫) ≥ 0 ,
(77)
everywhere in the entire space, for a given 𝜂(𝐫). As said before, if we let 𝜂(𝐫) to be
a density in , then 𝛽 must be in the range [0, 1] because of the convexity of .
Perdew and Levy proposed a kind of unconventional density variations (UDVs)
which satisfy [23]:
𝜂(𝐫)
√
𝜌 0 (𝐫)
∉ 
2
.
(78)
This class of UDVs gives rise to an unconventional energy variation, 𝛿E v , of subquadratic order of 𝛽 [25],
𝛿E v [𝜌 0 , 𝛿𝜌] = E v [𝜌 0 + 𝛿𝜌] − E v [𝜌 0 ] ∝ O(𝛽
b
), 1 < b < 2 ,
(79)
as opposed to the O(𝛽 2 ) behavior of conventional density variations (CDVs).