Functional Derivatives and Differentiability in Density-Functional Theory
359
We only need to further examine the situation when c 0 ≠ 0 as 𝛽 → 0 with the
choice of the negative sign in Eq. (143):
𝜆 = −2ac 0 − 𝜆
′
,
(144)
where the residual term 𝜆
′ approaches 0 as 𝛽 → 0:
𝜆
′
=
[
1
2ac 0
𝛽
2
−
1
8a 3 c
3
0
𝛽
4
+ ⋯
]
→ 0 .
(145)
Consequently, Eqs. (46) and (124) can be rewritten as
2a
[
c 0 (𝜌 0 − 𝜌 t ) + N

∑
i
c i Re
(⟨ Ψ 0
|
| Ψ i
⟩
N−1
)
]
= 𝜆
′
(
𝜌 t − 𝜌 
) ,
(146)
which immediately suggests that as 𝛽 → 0, (𝜌 0 − 𝜌 t ) and the coefficients {c i } for
Ψ i ∈  are linear in 𝜆
′ . Because 𝜌 t → 𝜌 0 , Ψ t → c 0 Ψ 0 with |c 0 | → 1, as 𝛽 → 0.
Therefore, at the end of the variational path (𝛽 = 0 and 𝜆 ′ = 0), 𝜆 = −2ac 0 , Ψ t =
c 0 Ψ 0 , |c 0 | = 1, and ̃
Ψ = −aΨ 0 .
Evidently, the choice of the negative sign in Eqs. (41) and (121) yields a fully
equivalent, alternative trial wavefunction,
̃
Ψ
′
= −aΨ 0 − 𝜆
′
Ψ t ,
(147)
where 𝜆 ′ → 0 as 𝛽 → 0. Then, we can carry out the discussion on the basis of 𝜆 ′ → 0
instead.
References
1. Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University
Press, New York
2. Hohenberg P, Kohn W (1964) Phys Rev 136:B864
3. Kohn W, Sham LJ (1965) Phys Rev 140:A1133
4. Wang YA, Xiang P (2013) In: Wesolowski TA, Wang YA (eds) Recent advances in orbital-free
density functional theory, Chap. 1. World Scientific, Singapore, pp 3–12
5. Lieb EH (1983) Int J Quantum Chem 24:243
6. Englisch H, Englisch R (1983) Phys Stat Sol 123:711
7. Englisch H, Englisch R (1984) Phys Stat Sol 124:373
8. Lindgren I, Salomonson S (2003) Phys Rev A 67:056501
9. Lindgren I, Salomonson S (2003) Adv Quantum Chem 43:95
10. Lindgren I, Salomonson S (2004) Phys Rev A 70:032509
11. Ekeland I, Temam R (1976) Convex analysis and variational problems. North-Holland, Amsterdam
12. Harris J, Jones RO (1974) J Phys F 4:1170