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P. Xiang and Y. A. Wang
the issue regarding the Gâteaux differentiability of density functionals [6, 7]. However, the Fréchet differentiability of density functionals remained unresolved until
Lindgren and Salomonson claimed its plausibility recently [8–10]. In this paper, we
reexamine Lindgren and Salomonson’s analysis to gain a better understanding about
the Fréchet differentiability of density functionals in DFT.
Mathematically speaking, a functional is a mapping from a function to a number.
G[f ], a functional of function f (x), can be expressed as f (x) ↦ G[f ]. The differential
of a functional, dG[f , 𝛿f ], is the part of the difference,
𝛿G[f , 𝛿f ] = G[f + 𝛿f ] − G[f ] ,
(1)
that depends on 𝛿f linearly:
dG[f , 𝛿f ] =
⟨ 𝛿G
𝛿f (x)
𝛿f (x)
⟩
,
(2)
where 𝛿G∕𝛿f (x) is the functional derivative of G[f ] with respect to f at point x. For
the sake of brevity, ⟨⋅⟩ is adopted as a shorthand notation for integration throughout
the text.
In DFT, there are two kinds of functional derivative: the Gâteaux derivative and
the Fréchet derivative [6–10]. Following Lindgren and Salomonson [8–10], all the
density functionals are defined on a convex space of densities,
 = {𝜌 | 𝜌 ≥ 0,
√ 𝜌 ∈ 
1
(
3
)} ,
(3)
where 
1
(
3
) is a Sobolev space [5]:

1
(
3
) = {q | q ∈ 
2
(
3
), 𝛁q ∈ 
2
(
3
)} ,
(4)
and  2 and  3 denote the spaces of square-integrable functions and threedimensional real coordinates, respectively. Rather than the general definitions given
in Appendix 1, we have slightly different definitions for functional differentiability
and functional derivatives [11].
Let G be a functional from  to the real numbers. If the limit
dG(𝜌 0 ; 𝛿𝜌) = lim
𝛽→0+
G(𝜌 0 + 𝛽𝛿𝜌) − G(𝜌 0 )
𝛽
(5)
exists, it is called the Gâteaux differential of G at 𝜌 0 in the direction 𝛿𝜌. If the limit
exists for any 𝛿𝜌 such that 𝜌 0 + 𝛽𝛿𝜌 ∈ , we say G is Gâteaux differentiable at 𝜌 0 .
Stated in another way, provided the functional G is Gâteaux differentiable at 𝜌 0 ,
the functional difference upon a density variation, 𝜌 0 (𝐫) → 𝜌 0 (𝐫) + 𝛽𝛿𝜌(𝐫), has two
terms,
𝛿G[𝜌 0 , 𝛽𝛿𝜌] = G[𝜌 0 + 𝛽𝛿𝜌] − G[𝜌 0 ] = 𝛽dG[𝜌 0 , 𝛿𝜌] + R[𝜌 0 , 𝛽𝛿𝜌] ,
(6)