Functional Derivatives and Differentiability in Density-Functional Theory
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where the last term satisfies the following limiting condition:
lim
𝛽→0+
R[𝜌 0 , 𝛽𝛿𝜌]
𝛽
= 0 ,
(7)
for a scaling parameter 0 ≤ 𝛽 ≤ 1. Usually we choose another density 𝜌
′
∈  to
be 𝛿𝜌 and require 𝜌 = 𝜌 0 + 𝛽𝜌 ′ to be always in  during the variational process;
it follows immediately from the definition of a convex subspace (see Appendix 1)
that the range of 𝛽 must be [0, 1]. In a rigorous sense, the differential dG needs to
be neither linear nor continuous in 𝛿𝜌. However, most applied literature adopt the
convention that dG is linear and continuous in 𝛿𝜌. Consequently, dG can be written
as
dG[𝜌 0 , 𝛿𝜌] =
⟨ 𝛿G
𝛿𝜌 0 (𝐫)
𝛿𝜌(𝐫)
⟩
,
(8)
in which 𝛿G∕𝛿𝜌 0 (𝐫) is called the Gâteaux derivative of G at 𝜌 0 (𝐫). Similar to a directional derivative, the Gâteaux derivative is a functional of 𝜌 0 (𝐫) only, although along
various directions it might take different expressions (e.g., different functions of the
variable 𝐫).
If the last term of the functional difference,
𝛿G[𝜌 0 , 𝛿𝜌] = G[𝜌 0 + 𝛿𝜌] − G[𝜌 0 ] =
⟨ 𝛿G
𝛿𝜌 0 (𝐫)
𝛿𝜌(𝐫)
⟩
+ R[𝜌 0 , 𝛿𝜌] ,
(9)
instead satisfies
lim
||𝛿𝜌||→0
R[𝜌 0 , 𝛿𝜌]
||𝛿𝜌||
= 0 ,
(10)
for the norm of 𝛿𝜌(𝐫), ||𝛿𝜌|| = ⟨|𝛿𝜌(𝐫)|⟩, 𝛿G∕𝛿𝜌 0 (𝐫) is called the Fréchet derivative.
The Fréchet derivative is a global derivative: all directions approaching 𝜌 0 (𝐫) yield
the same derivative (with the same expression). By default, Fréchet differentiability
is stronger than Gâteaux differentiability.
2 Controversy over Fréchet Differentiability
Consider an N-electron quantum system under the influence of a local electronnuclear potential v(𝐫). In the adiabatic connection formulation [12–17], the total
Hamiltonian operator ̂
H is a sum of three terms, the kinetic-energy operator ̂
T, the
potential-energy operator ̂
V, and the inter-electron coulombic repulsion operator ̂
W
with an adiabatic connection parameter 𝜔:
̂
H = ̂
T + 𝜔 ̂
W + ̂
V = −
1
2
N
∑
i=1
∇
2
i + 𝜔
N
∑
i 1
|𝐫 i − 𝐫 j |
+
N
∑
i=1
v(𝐫 i ) ,
(11)