478
N. C. Bacalis
F n [φ 0 , φ 1 , . . . ; ψ n ] = E n ,
i.e., ψ n is a critical point, the saddle point, regardless of the quality of φ i < n .
Incidentally, notice that, since E n is a saddle point, the energy E[φ n ] of an
approximant φ n is not necessarily an upper bound of E n ; it must only be an upper
bound of (E n – L): Since E[φ n ] + L − E n = U > 0, then E n − L < E[φ n ], or
E n −
i
2 < E [φ n ] ,
(13.8)
i.e., it may be below E n (that is why the aforementioned “orthogonal” minimization
(OO) does not “collapse”). This means that since, in practice, with a truncated
expansion, the minimization procedure of F n , within a tolerance, stops around
(but not exactly at) ψ n , if it stops at a side of upward parabola, we will have
E n < E[φ n ] < F n , but if it stops at a side of downward parabola, we will have E n –
L < E[φ n ] < E n < F n . In this case, the final (converged) φ n , although slightly below
E[HUM] (by ≤ L), will be certainly acceptable. The value of L could be estimated
using accurate approximants φ i < n and checked to be of the order of the tolerance.
Further, at φ n = ψ n , if we denote by
φ
⊥{n}
i
the projection of |φ i on the subspace
of the higher than-n eigenfunctions, the Hessian determinant is
A n
n =2 n+1
n−1
i=0
(E n −E i )
E
φ
⊥{n}
n
−E n
×
×
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1+2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
n
n−1
i=0
n|φ i 2 +
+
n−1
i=0
E
φ
⊥{n}
n
−E n
E
φ
⊥{n}
i
−E n
−
φ
⊥{n}
i
H −E n
φ
⊥{n}
n
2
(E n −E i )
E
φ
⊥{n}
i
−E n
φ
⊥{n}
i
|φ i
2
− 2
n−1
i=0
n−1
j =i+1
((En−Ej )j|φi+(En−Ei)i|φj )
2
(E n −E i )(E n −E j )
+ O[coefficients]
3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
(13.9)
and its principal minors are
A k
= 2 k+1
k
i=0
(E n − E i ) ×
×
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1 + 2
⎡
⎢
⎢
⎢
⎣
(k + 1)
n−1
i=0
n|φ i 2 +
n−1
j =k+1
k
i=0
En−E i
En−E j
i|φ j
2 +
k
i=0
E
φ
⊥{n}
i
−En
En−E i
φ
⊥{n}
i
|φ i
2
−
n−1
j =k+1
k
i=0
En−E j
En−E i
j |φ i 2 − 2
k
i=0
k
j =i+1
((En−Ej )j|φi+(En−Ei)i|φj )
2
(En−Ei )(En−Ej )
⎤
⎥
⎥
⎥
⎦
+ O[coefficients]
3 .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(13.10)
N. C. Bacalis
F n [φ 0 , φ 1 , . . . ; ψ n ] = E n ,
i.e., ψ n is a critical point, the saddle point, regardless of the quality of φ i < n .
Incidentally, notice that, since E n is a saddle point, the energy E[φ n ] of an
approximant φ n is not necessarily an upper bound of E n ; it must only be an upper
bound of (E n – L): Since E[φ n ] + L − E n = U > 0, then E n − L < E[φ n ], or
E n −
i
2 < E [φ n ] ,
(13.8)
i.e., it may be below E n (that is why the aforementioned “orthogonal” minimization
(OO) does not “collapse”). This means that since, in practice, with a truncated
expansion, the minimization procedure of F n , within a tolerance, stops around
(but not exactly at) ψ n , if it stops at a side of upward parabola, we will have
E n < E[φ n ] < F n , but if it stops at a side of downward parabola, we will have E n –
L < E[φ n ] < E n < F n . In this case, the final (converged) φ n , although slightly below
E[HUM] (by ≤ L), will be certainly acceptable. The value of L could be estimated
using accurate approximants φ i < n and checked to be of the order of the tolerance.
Further, at φ n = ψ n , if we denote by
φ
⊥{n}
i
the projection of |φ i on the subspace
of the higher than-n eigenfunctions, the Hessian determinant is
A n
n =2 n+1
n−1
i=0
(E n −E i )
E
φ
⊥{n}
n
−E n
×
×
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1+2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
n
n−1
i=0
n|φ i 2 +
+
n−1
i=0
E
φ
⊥{n}
n
−E n
E
φ
⊥{n}
i
−E n
−
φ
⊥{n}
i
H −E n
φ
⊥{n}
n
2
(E n −E i )
E
φ
⊥{n}
i
−E n
φ
⊥{n}
i
|φ i
2
− 2
n−1
i=0
n−1
j =i+1
((En−Ej )j|φi+(En−Ei)i|φj )
2
(E n −E i )(E n −E j )
+ O[coefficients]
3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
(13.9)
and its principal minors are
A k
= 2 k+1
k
i=0
(E n − E i ) ×
×
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1 + 2
⎡
⎢
⎢
⎢
⎣
(k + 1)
n−1
i=0
n|φ i 2 +
n−1
j =k+1
k
i=0
En−E i
En−E j
i|φ j
2 +
k
i=0
E
φ
⊥{n}
i
−En
En−E i
φ
⊥{n}
i
|φ i
2
−
n−1
j =k+1
k
i=0
En−E j
En−E i
j |φ i 2 − 2
k
i=0
k
j =i+1
((En−Ej )j|φi+(En−Ei)i|φj )
2
(En−Ei )(En−Ej )
⎤
⎥
⎥
⎥
⎦
+ O[coefficients]
3 .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(13.10)