356
P. Xiang and Y. A. Wang
𝐁 = 2𝜆
√
1 − 𝛽 2
[
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
)
,
(130)
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⟩ ,
(131)
where h and g(𝐫) are the Lagrange multipliers corresponding to the two constraints
in Eqs. (129) and (130). Minimizing Eq. (131) with respect to {c i }, one obtains
𝜆
⟨[ √
1 − 𝛽 2 (𝜌 0 − 𝜌  )
N
+ 𝜆
∞
∑
j=0
c j Re
( ⟨ Ψ 0 |
| Ψ j
⟩
N−1
)
]
g(𝐫)
⟩
= −hc 0 ,
(132)
and
𝜆
⟨[
√
1 − 𝛽 2 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) .
(133)
Because c 0 is linear in 𝜆 as we previously showed, we can readily infer from Eq.
(132) that g(𝐫) must take the following form:
g 𝜆 (𝐫) = g
(0)
(𝐫) +
∞
∑
k=1
g
(k)
(𝐫)
k!
𝜆
k
.
(134)
Substituting Eq. (134) into Eq. (133) and ignoring the higher-order terms as 𝜆 → 0,
we obtain an equation for Ψ i ∈ ,
−hc i = 𝜆
√
1 − 𝛽 2
⟨
Re
(⟨ Ψ i |
| Ψ 0
⟩
N−1
)
g
(0)
(𝐫)
⟩ + h.o. ,
(135)