Functional Derivatives and Differentiability in Density-Functional Theory
337
where N is the total number of variables and m is the number of variables excluded
in the integration. If f is chosen to be the seed function, we can define the order-m
SOF space that consists all functions of order-m strongly orthogonal to the same seed
function f .
Because spin operator commutes with the Hamiltonian, they share a common
complete set of eigenfunctions. In the forthcoming discussion, we only consider
wavefunctions in Hilbert space spanned by this complete set. Let the seed function
be the N-electron GS wavefunction Ψ 0 and the integration in Eq. (29) be over all spatial and spin coordinates except m spatial coordinates, and label  as its associated
order-1 SOF space and  as its order-0 SOF space with  excluded. For a general
Hamiltonian (0 ≤ 𝜔 ≤ 1), there is no doubt that  exists. Since we integrate over
all spin coordinates, if any non-GS eigenfunction Ψ has a different spin multiplicity
from that of Ψ 0 , we have
⟨Ψ 0 |Ψ⟩ N−1 = 0 ,
(30)
then Ψ is in . Therefore,  does exist in general. For the noninteracting system
(𝜔 = 0) in particular, Ψ 0 is a single Slater determinant,  includes all singly excited
determinants, and  has all doubly and higher excited determinants. Thus, the complete set of eigenfunctions {Ψ i } of the Hamiltonian can be partitioned into three
parts: Ψ 0 , , and .
Let a normalized wavefunction Ψ  be a linear combination of eigenfunctions in
. Hereafter, we are going to show that along one particular variational path defined
by
Ψ p = Ψ 0 + í µí»½Ψ  ,
(31)
the Fréchet derivative does not exist. For the wavefunction Ψ p , its corresponding
electron density is
𝜌 p (𝐫) = N
⟨
Ψ p
|
|
|
Ψ p
⟩
N−1
= N
⟨ Ψ 0
|
| Ψ 0
⟩
N−1
+ 𝛽
2 N
⟨ Ψ 
|
| Ψ 
⟩
N−1
+ 2𝛽NRe
(⟨ Ψ 
|
| Ψ 0
⟩
N−1
)
= 𝜌 0 (𝐫) + 𝛽
2
𝜌  (𝐫) ,
(32)
where Re
( ⟨Ψ  |Ψ 0 ⟩ N−1
)
, the real part of ⟨Ψ  |Ψ 0 ⟩ N−1 , is zero because of the nature
of Ψ  . When 𝛽 approaches 0, 𝜌 p (𝐫) also approaches 𝜌 0 (𝐫). Clearly, 𝜌 p (𝐫) lies in the
neighborhood of 𝜌 0 (𝐫) within . Equation (32) specifies the density variation path
for the forthcoming discussion. For later convenience, we label  as the set of all
legitimate 𝜌 p (𝐫) for a given Ψ 0 or 𝜌 0 (𝐫). Throughout the text, Re(⋅) will be used to
denote the real part of the quantity involved.
Along this particular variational path, let ̃
Ψ be a trial wavefunction yielding the
same density 𝜌 p (𝐫):
̃
Ψ = Ψ 0 + í µí¼†Ψ t ⟼ 𝜌 p (𝐫) ,
(33)