Functional Derivatives and Differentiability in Density-Functional Theory
347
where K(𝐫) has no dependence on 𝛿𝜌 ′ (𝐫) and R[𝜌 0 , 𝛿𝜌 ′ ] satisfies:
lim
𝛽→0
R[𝜌 0 , 𝛿𝜌 ′ ]
𝛽
= 0 .
(85)
Substituting Eq. (84) into Eq. (83), we obtain
𝛿F
𝜔
LL =
⟨ 𝛿F
𝜔
LL
𝛿𝜌(𝐫)
|
|
|
|
|𝜌 0
𝛿𝜌
′
(𝐫)
⟩
+ R[𝜌 0 , 𝛿𝜌
′
] ,
(86)
where the new Gâteaux derivative is
𝛿F
𝜔
LL
𝛿𝜌(𝐫)
|
|
|
|
|𝜌 0
=
E 0
N
− v(𝐫) + K(𝐫) .
(87)
If a CDV path is chosen instead, we will get the Gâteaux derivative previously
obtained by Lindgren and Salomonson [8–10]:
𝛿F
𝜔
LL
𝛿𝜌(𝐫)
|
|
|
|
|𝜌 0
=
E 0
N
− v(𝐫) .
(88)
Equations (87) and (88) clearly indicate that the nonuniqueness of the Gâteaux
derivative along different paths. For the Lieb functionals F
𝜔
L
, it can be easily shown
that the same conclusion is still valid.
Because of the existence of UDVs of the first kind [23] and the second kind,
density variation domain has to be cleansed so that consistent results can be obtained.
However, there remains one problem: What is the relationship between these two
kinds of UDVs? In other words, does the set of Perdew and Levy’s UDVs contains
our UDVs or is the opposite true? Consider a noninteracting two-electron atom with
nuclear charge Z = 2, its ground state is 1s 2 and the GS wavefunction is denoted by
Φ 1s 2 . If both electrons are excited to the 2s orbital, we would get a wavefunction Φ 2s 2
from the  space of Φ 1s 2 . The density variation,
𝜌(𝐫) = 𝜌 0 (𝐫) + 𝛽𝛿𝜌(𝐫) = 𝜌 1s 2 (𝐫) + 𝛽𝜌 2s 2 (𝐫) ,
(89)
with 0 ≤ 𝛽 ≤ 1, would belong to the UDVs of the second kind. Due to the asymptotic
behavior of wavefunctions [1], we have
lim
𝐫→∞
𝜌 1s 2 (𝐫) ∼ e
−2𝐫
√ 2I 1s 2
(90)
and
lim
𝐫→∞
𝜌 2s 2 (𝐫) ∼ e
−2𝐫
√ 2I 2s 2 ,
(91)