Functional Derivatives and Differentiability in Density-Functional Theory
353
with
lim
||s||  →0
||𝜖(x; s)|| 
||s|| 
= 0 .
(112)
The operator T
′
F
(x) is called the Fréchet or strong derivative of T at x. The Fréchet
derivative at x is unique. It can be shown that the existence of the Fréchet derivative
of T at x implies continuity of T at x.
Theorem 1 If the Gâteaux derivative T
′
G
(x) exists in the neighborhood of x and is
continuous with respect to the norm in (,  ) at x, then the Fréchet derivative
T
′
F
(x) exists and is equal to T
′
G
(x).
Appendix 2
In this appendix, we show that the Fréchet derivative does not exist in the normalized
density domain,  N .
Define a normalized path wavefunction,
Ψ p =
√
1 − 𝛽 2 Ψ 0 + í µí»½Ψ  ,
(113)
where Ψ 0 is the GS wavefunction for an N-electron quantum system, Ψ  is a linear combination of eigenfunctions in  of Ψ 0 , and 0 ≤ 𝛽 ≤ 1. Both Ψ 0 and Ψ  are
normalized to 1. The corresponding path density is
𝜌 p (𝐫) = N⟨Ψ p |Ψ p ⟩ N−1
= (1 − 𝛽
2
)N⟨Ψ 0 |Ψ 0 ⟩ N−1 + 𝛽
2 N⟨Ψ  |Ψ  ⟩ N−1
= (1 − 𝛽
2
)𝜌 0 (𝐫) + 𝛽
2
𝜌  (𝐫) .
(114)
When 𝛽 approaches 0, 𝜌 p (𝐫) also approaches 𝜌 0 (𝐫). Letting 𝛽 changes continuously
from 1 to 0, we obtain the desired density variational path. Equation (114) shows
that the path density is automatically normalized to N, therefore the density variation
stays within the normalized space. Clearly, 𝜌 p (𝐫) lies in the neighborhood of 𝜌 0 (𝐫)
within  N . For convenience, we label  N as the set of all legitimate N-representable
𝜌 p (𝐫) defined for a given Ψ 0 or 𝜌 0 (𝐫) in Eq. (114).
A trial wavefunction is then assumed to yield the same path density:
̃
Ψ =
√
1 − 𝛽 2 Ψ 0 + í µí¼†Ψ t =
√
1 − 𝛽 2 Ψ 0 + 𝜆
∞
∑
i=0
c i Ψ i ⟼ 𝜌 p (𝐫) ,
(115)
where Ψ i is the ith normalized eigenfunction of ̂
H, ⟨Ψ t |Ψ t ⟩ = 1, and the expansion
coefficients {c i } are chosen to be real. The complete set of {Ψ i } can be divided into
three parts: Ψ 0 , , and . The electron density (the trial density) for ̃
Ψ takes the