346
P. Xiang and Y. A. Wang
In general, the energy variation can be written as
𝛿E v [𝜌 0 , 𝛿𝜌] =
⟨ 𝛿E v
𝛿𝜌(𝐫)
|
|
|
|𝜌 0
𝛿𝜌(𝐫)
⟩
+ h.o. ,
(80)
where “h.o.” encompasses all higher-order terms in 𝛿𝜌(𝐫). Zhang and Wang [25]
recently discovered that Eq. (80) cannot be expanded to second order in 𝛿𝜌(𝐫) for
the UDVs of the first kind [23]. In other words, the second (or any higher-order)
functional derivative of the total energy functional may not exist for all allowed density variations within the current DFT framework. This imposes a serious problem
in DFT since the effort to interpret the second functional derivative as the chemical
hardness [1] will fail for the UDVs of the first kind [23]. Some modification of the
density variation domain must be in place to rescue the situation.
From a different perspective, we introduce a new definition of UDV based on
our discussion in Sect. 3. If a density variation, 𝛿𝜌(𝐫) = 𝜌(𝐫) − 𝜌 0 (𝐫), comes from a
density 𝜌(𝐫) ∈ , then the density variation is an UDV. For these UDVs of the second
kind, their corresponding Gâteaux derivatives are different from the conventional
one [8–10], or simply, not Fréchet derivatives.
Let us write the wavefunction in Eq. (31) differently as Ψ ′
p = Ψ 0 +
√
í µí»½Ψ  , the
associated density variation will become linear in 𝛽,
𝛿𝜌
′
(𝐫) = 𝛽𝜌  (𝐫) .
(81)
Because of Eqs. (31), (33), and (62), í µí»¿Ψ ′ =
√ í µí»½Ψ t , and Eq. (65) is invariant,
inf
Ψ 0 +í µí»¿Ψ ′ →𝜌 0 +𝛿𝜌 ′
⟨í µí»¿Ψ ′ | ̂
H − E 0 |í µí»¿Ψ ′ ⟩
||𝛿𝜌 ′ ||
= inf
Ψ t →𝜌 
⟨
√
í µí»½Ψ t | ̂
H − E 0 |
√
í µí»½Ψ t ⟩
||𝛽𝜌  (𝐫)||
=
inf
Ψ t →𝜌 
⟨Ψ t | ̂
H − E 0 |Ψ t ⟩
||𝜌  ||
= constant ≠ 0 .
(82)
Equation (25) then becomes
𝛿F
𝜔
LL =
⟨[ E 0
N
− v(𝐫)
]
𝛿𝜌
′
(𝐫)
⟩
+
inf
Ψ 0 +í µí»¿Ψ ′ →𝜌 0 +𝛿𝜌 ′
⟨𝛿Ψ
′
| ̂
H − E 0 |𝛿Ψ
′
⟩ .
(83)
From Eq. (82), we know that the second term on the RHS of Eq. (83) is of linear
order in 𝛿𝜌 ′ (𝐫) when the density is approaching 𝜌 0 (𝐫). We can then split the second
term in Eq. (83) into a first-order term of 𝛿𝜌
′
(𝐫) and a higher-order residual:
inf
Ψ 0 +í µí»¿Ψ ′ →𝜌 0 +𝛿𝜌 ′
⟨𝛿Ψ
′
| ̂
H − E 0 |𝛿Ψ
′
⟩ = ⟨K(𝐫)𝛿𝜌
′
(𝐫)⟩ + R[𝜌 0 , 𝛿𝜌
′
] ,
(84)