Functional Derivatives and Differentiability in Density-Functional Theory
335
 N ≡ {𝜌 | 𝜌 ≥ 0, ⟨𝜌⟩ = N,
√ 𝜌 ∈ 
1
(
3
)} .
(19)
If 𝜔 = 0, Eq. (18) reduces to the definition of the noninteracting kinetic-energy density functional,
T s [𝜌] = inf
Ψ→𝜌
⟨Ψ| ̂
T|Ψ⟩.
(20)
Built upon the following paradoxical statement [4]:
𝛿T s [𝜌]
𝛿𝜌(𝐫)
= 𝜀 i − v eff (𝐫),
(21)
Nesbet asserted that the functional derivative of the noninteracting kinetic-energy
functional in unnormalized density domain is a Gâteaux derivative rather than a
Fréchet derivative [19, 20]. Shortly after, Lindgren and Salomonson [8–10] refuted
Nesbet by pointing out that the noninteracting kinetic-energy functional used by Nesbet was not a proper density functional and further reasoned the Fréchet differentiability of the noninteracting kinetic-energy functional.
Based on Eqs. (16) and (17), Lindgren and Salomonson [8–10] obtained the
expression of the HK universal density functional for the GS density 𝜌 0 (𝐫) and wavefunction Ψ 0 ,
F
𝜔
[𝜌 0 ] = ⟨Ψ 0 | ̂
F
𝜔
|Ψ 0 ⟩ = ⟨Ψ 0 | ̂
H − ̂
V|Ψ 0 ⟩ =
⟨[ E 0
N
− v(𝐫)
]
𝜌 0 (𝐫)
⟩
,
(22)
where E 0 is the GS energy. Because of the identity for arbitrary Ψ = Ψ 0 + 𝛿Ψ,
⟨Ψ| ̂
F
𝜔
|Ψ⟩ = ⟨Ψ| ̂
H|Ψ⟩ − ⟨v(𝐫) 𝜌(𝐫)⟩
= E 0 ⟨Ψ|Ψ⟩ + ⟨𝛿Ψ| ̂
H − E 0 |𝛿Ψ⟩ − ⟨v(𝐫) 𝜌(𝐫)⟩
=
⟨[ E 0
N
− v(𝐫)
]
𝜌(𝐫)
⟩
+ ⟨𝛿Ψ| ̂
H − E 0 |𝛿Ψ⟩ ,
(23)
the corresponding Levy-Lieb density functional takes the following form:
F
𝜔
LL [𝜌 0 + 𝛿𝜌] = inf
Ψ→𝜌 0 +𝛿𝜌
⟨Ψ| ̂
F
𝜔
|Ψ⟩
=
⟨[ E 0
N
− v(𝐫)
] [ 𝜌 0 (𝐫) + 𝛿𝜌(𝐫)
]
⟩
+
inf
Ψ 0 +𝛿Ψ→𝜌 0 +𝛿𝜌
⟨𝛿Ψ| ̂
H − E 0 |𝛿Ψ⟩ . (24)
Here, the GS density, 𝜌 0 , is a pure-state v-representable (PS-v-representable) density,
which corresponds to a single GS wavefunction of a Hamiltonian with a physically
reasonable potential v in the space of  ∞ +  3∕2 [5, 9, 21]. Subtracting Eq. (22)
from Eq. (24) leads to