Functional Derivatives and Differentiability in Density-Functional Theory
341
𝐁 = 2𝜆
[
c 0 (𝜌 0 − 𝜌  )
N
+

∑
i
c i Re
(⟨ Ψ 0 |
| Ψ i
⟩
N−1
)
]
−𝜆
2
(
𝜌 
N
−
∞
∑
i,j
c i c j
⟨ Ψ i |
| Ψ j
⟩
N−1
)
,
(53)
and
𝛀 = 𝜆
2
(
⟨Ψ t | ̂
H|Ψ t ⟩ − E 0
) − h𝐀 − ⟨g(𝐫) 𝐁⟩
= 𝜆
2
( ∞
∑
i,j
c i c j ⟨Ψ i | ̂
H|Ψ j ⟩ − E 0
)
− h𝐀 − ⟨g(𝐫) 𝐁⟩
= 𝜆
2
( ∞
∑
i,j
c i c j E j 𝛿 ij − E 0
)
− h𝐀 − ⟨g(𝐫) 𝐁⟩
= 𝜆
2
[ ∞
∑
i=1
c
2
i
(E i − E 0 )
]
− h𝐀 − ⟨g(𝐫) 𝐁⟩ ,
(54)
where h and g(𝐫) are the Lagrange multipliers corresponding to the two constraints
in Eqs. (52) and (53), and Eq. (49) has been used to derive the last expression. Minimizing Eq. (54) with respect to {c i }, one obtains
𝜆
⟨[
𝜌 0 − 𝜌 
N
+ 𝜆
∞
∑
j=0
c j Re
( ⟨ Ψ 0 |
| Ψ j
⟩
N−1
)
]
g(𝐫)
⟩
= −hc 0 ,
(55)
and
𝜆
⟨[
Re
(⟨ Ψ i
|
| Ψ 0
⟩
N−1
) + 𝜆
∞
∑
j=0
c j Re
( ⟨ Ψ i
|
| Ψ j
⟩
N−1
)
]
g(𝐫)
⟩
=
[ 𝜆
2
(E i − E 0 ) − h
]
c i
(for i ≠ 0) .
(56)
Because c 0 is at least linear in 𝜆 as we showed above, we can readily infer from Eq.
(55) that g(𝐫) must take the following form,
g 𝜆 (𝐫) = g
(0)
(𝐫) +
∞
∑
k=1
g (k) (𝐫)
k!
𝜆
k
,
(57)
where g
(0) can be zero depending on whether c 0 is higher-than-linear in 𝜆 or not.
Substituting Eq. (57) back into Eq. (56) and ignoring the higher-order terms when 𝜆
approaches 0, we obtain a much simplified expression of c i for Ψ i ∈ ,
−hc i = 𝜆
⟨
Re
(⟨ Ψ i
|
| Ψ 0
⟩
N−1
)
g
(0)
(𝐫)
⟩ + h.o. ,
(58)