Functional Derivatives and Differentiability in Density-Functional Theory
339
Because both 𝜆 and 𝛽 approach 0 concurrently near the end of the variational path,
only the positive sign (see Appendix 3) is allowed in Eq. (41). Thus, as 𝛽 → 0, we
get
𝜆 =
1
2c 0
𝛽
2
−
1
8c
3
0
𝛽
4
+ ⋯ .
(42)
Immediately, we can conclude that towards the end of the variational path, 𝜆 is of
the same magnitude as 𝛽
2
∕c 0 .
If c 0 = 0, Eq. (39) immediately reduces to 𝜆
2
= 𝛽
2 . In this case, the wavefunction
variation í µí»¿Ψ in Eq. (26) can be regarded as linear in 𝛽 and the density variation 𝛿𝜌 =
𝛽
2
𝜌  is quadratic in 𝛽 as shown by Eq. (32). This immediately invalidates Fréchet
differentiability by destroying Eq. (26). Nonetheless, we are able to gain much deeper
understanding about the structure of Ψ t through the following analysis.
Consider the case when c 0 ≠ 0. Because of Eqs. (32), (35), and (36), we obtain
𝜆
2
𝜌 t + 2𝜆NRe
(⟨ Ψ 0 |
| Ψ t
⟩
N−1
) = 𝛽
2
𝜌  .
(43)
Substituting Eq. (39) into Eq. (43) and grouping terms according to the powers of 𝜆,
we have
2𝜆
[
c 0 𝜌  − N
 0
∑
i
c i Re
(⟨ Ψ 0
|
| Ψ i
⟩
N−1
)
]
= 𝜆
2
(𝜌 t − 𝜌  ) ,
(44)
where the summation on the left-hand side (LHS) is only within the combined space
of  and Ψ 0 ,
 0 ≡  ∪ Ψ 0 .
(45)
Separating the Ψ 0 contribution from the summation, we can simplify Eq. (44) to
2c 0 (𝜌 0 − 𝜌  ) + 2N

∑
i
c i Re
(⟨ Ψ 0 |
| Ψ i
⟩
N−1
) = 𝜆
( 𝜌  − 𝜌 t
) .
(46)
Because 𝜌 0 , 𝜌  , and all Ψ i in Eq. (46) have no dependence on 𝜆, we must admit that
c i corresponding to any wavefunction in  0 must be a function of 𝜆.
Based on perturbation theory, c i (𝜆) can be expanded as
c i (𝜆) = c
(0)
i
+ c
(1)
i
𝜆 + h.o. ,
(47)
where “h.o.” represent higher-order terms in 𝜆. It can be readily shown that all the
c
(0)
i
must be zero in order to satisfy Eq. (46) for any 𝜆 → 0, whereas c
(1)
i
can be zero
or non-zero. Consequently, all {c i } for Ψ i in  0 vary at least linearly in 𝜆. Some c i