338
P. Xiang and Y. A. Wang
where Ψ t is normalized to 1 and 𝜆 is a real scaling parameter (potentially different
from 𝛽). We can expand Ψ t in terms of the complete set of normalized eigenfunctions
{Ψ i } of ̂
H,
Ψ t =
∞
∑
i=0
c i Ψ i ,
(34)
where c i is the expansion coefficient of Ψ i . Without loss of generality, we can choose
all expansion coefficients {c i } to be real because any complex phases in {c i } can be
attributed to the corresponding {Ψ i } instead. The electron density for ̃
Ψ then takes
the following form:
̃
𝜌(𝐫) = N⟨ ̃
Ψ| ̃
Ψ⟩ N−1
= N⟨Ψ 0 |Ψ 0 ⟩ N−1 + 𝜆
2 N⟨Ψ t |Ψ t ⟩ N−1 + 2𝜆NRe
( ⟨Ψ 0 |Ψ t ⟩ N−1
)
= 𝜌 0 (𝐫) + 𝜆
2
𝜌 t (𝐫) + 2𝜆NRe
( ⟨Ψ 0 |Ψ t ⟩ N−1
) .
(35)
Clearly, as 𝜆 → 0, ̃
𝜌(𝐫) also approaches 𝜌 0 (𝐫).
Because the density variation is along the path with the same density (but with
different wavefunctions),
̃
𝜌(𝐫) = 𝜌 p (𝐫) → 𝜌 0 (𝐫) ,
(36)
the normalization of the two densities must be identical,
⟨̃ 𝜌(𝐫)⟩ =
⟨
𝜌 p (𝐫)
⟩ .
(37)
Substituting Eqs. (32) and (35) into Eq. (37), one has
𝜆
2
⟨𝜌 t (𝐫)⟩ + 2𝜆NRe
(⟨ Ψ 0 |
| Ψ t
⟩) = 𝛽
2
⟨𝜌  (𝐫)⟩ .
(38)
Since ⟨𝜌  (𝐫)⟩ and ⟨𝜌 t (𝐫)⟩ are all equal to N, Eq. (38) can be readily simplified to
𝛽
2
= 𝜆
2
+ 2𝜆Re
(⟨ Ψ 0
|
| Ψ t
⟩) = 𝜆
2
+ 2𝜆
∞
∑
i=0
Re
(
c i
⟨ Ψ 0
|
| Ψ i
⟩) = 𝜆
2
+ 2𝜆c 0 . (39)
At any specific point along the variational path, the value of 𝛽 is fixed, and we can
solve 𝜆 in terms of 𝛽 based on Eq. (39):
𝜆 = −c 0 ±
√
c
2
0
+ 𝛽 2 .
(40)
If c 0 ≠ 0 when 𝛽 approaches 0, 𝜆 takes a Taylor-series expansion,
𝜆 = −c 0 ±
[
c 0 +
1
2c 0
𝛽
2
−
1
8c
3
0
𝛽
4
+ ⋯
]
.
(41)