340
P. Xiang and Y. A. Wang
may have a higher-than-linear dependence on 𝜆, but at least one c i is linear in 𝜆 for
Eq. (46) to be valid. At this point, we do not know the exact behavior of those {c i } for
Ψ i in . Actually, in the above derivation, we only considered one constraint along
the variational path, namely, Eq. (36), but ignored the other constraint, ⟨Ψ t |Ψ t ⟩ = 1.
What will happen if both constraints are enforced concurrently?
By the minimization process, we can find the best set of coefficients {c i } that
delivers the lowest energy out of the following energy functional for the given 𝜌 p (𝐫)
with a fixed 𝛽,
⟨ ̃
Ψ| ̂
H| ̃
Ψ⟩ = ⟨Ψ 0 + í µí¼†Ψ t | ̂
H|Ψ 0 + í µí¼†Ψ t ⟩
= ⟨Ψ 0 + í µí¼†Ψ t | ̂
H|Ψ 0 ⟩ + ⟨Ψ 0 + í µí¼†Ψ t | ̂
H|í µí¼†Ψ t ⟩
= E 0 ⟨ ̃
Ψ|Ψ 0 ⟩ + ⟨Ψ 0 | ̂
H|í µí¼†Ψ t ⟩ + 𝜆
2
⟨Ψ t | ̂
H|Ψ t ⟩
= E 0 ⟨ ̃
Ψ| ̃
Ψ⟩ − E 0 ⟨ ̃
Ψ|í µí¼†Ψ t ⟩ + E 0 ⟨Ψ 0 |í µí¼†Ψ t ⟩ + 𝜆
2
⟨Ψ t | ̂
H|Ψ t ⟩
= E 0 ⟨ ̃
Ψ| ̃
Ψ⟩ − 𝜆
2 E 0 ⟨Ψ t |Ψ t ⟩ + 𝜆
2
⟨Ψ t | ̂
H|Ψ t ⟩
= (1 + 𝛽
2
)E 0 + 𝜆
2
( ⟨Ψ t | ̂
H|Ψ t ⟩ − E 0
) .
(48)
To achieve this task, one only needs to minimize 𝜆 2 (
⟨Ψ t | ̂
H|Ψ t ⟩ − E 0
)
in Eq. (48)
under the following two constraints:
∞
∑
i=0
c
2
i = 1 ,
(49)
and
̃
𝜌(𝐫) = 𝜌 p (𝐫) .
(50)
The second constraint, Eq. (50), is equivalent to the following equation based on our
previous analysis:
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
)
.
(51)
We will use the Euler-Lagrange multiplier method to find the set of coefficients {c i }
that minimizes the value of 𝜆 2 (
⟨Ψ t | ̂
H|Ψ t ⟩ − E 0
)
.
Define
𝐀 =
∞
∑
i=0
c
2
i − 1 ,
(52)