480
N. C. Bacalis
with energy
E
φ
+
0
= E [φ 0 ] −
(E [ψ 1 ] − E [φ 0 ]) ψ 1 |φ 0 2
1 − ψ 1 |φ 0 2
≤ E [φ 0 ]
(13.13)
(i.e., − is same or better than φ 0 ). Further, rotating φ
+
0 around ψ 1 would
improve φ
+
0 as follows: Introduce (e.g., by one more configuration) a function φ
(2+)
0
orthogonal to both {φ
+
0 , ψ 1 }. Then, in the subspace of {φ
+
0 , φ
(2+)
0
}, (both orthogonal
to ψ 1 ), the Hamiltonian opens their energy gap, so the lowest eigenvector − ≡ φ
−
0
has energy E
φ
−
0
≤ E
φ
+
0
, closer to E[ψ 0 ] (in a three-dimensional function
space {ψ 0 , ψ 1 , ψ k }, this would be exactly E[ψ 0 ] as directly verified). E
φ
−
0
could
be further improved by further rotating around ψ 1 similarly: i.e., after introducing
another function φ
(3+)
0
orthogonal to both {φ
−
0 , ψ 1 } in the subspace of {φ
−
0 , φ
(3+)
0
}
(both orthogonal to ψ 1 ), the lowest Hamiltonian eigenvector is − ≡ φ
(2−)
0
with
energy E
φ
(2−)
0
≤ E
φ
−
0
(even closer to E[ψ 0 ]) and so on.
Now let us consider the approximants.
13.4.2 Improving φ 0 Orthogonally to φ 1
Since ψ 1 is never exactly known, we can still improve φ 0 orthogonally to φ 1 , our
best approximant of ψ 1 (obtained either via F 1 or via a “large” HUM expansion),
by first computing φ
+
0 orthogonal to our φ 1 ,
φ
+
0 ≡
φ 0 − φ 1 φ 1 |φ 0
1 − φ 1 |φ 0 2
,
(13.14)
if the condition E
φ
+
0
≤ E [φ 0 ] is attainable: Indeed, by expanding about
φ 1 , as directly verified, the condition E
φ
+
0
≤ E [φ 0 ], to leading order, reads
(E 1 − E 0 )
1 − ψ 1 |φ 0 2 ≥
E
φ
⊥{1}
0
− E 0
φ
⊥{1}
0
|φ 0
2
, which is not impossible. Here, as defined in Eqs. (13.9) and (13.10), φ
⊥{1}
0
is the normalized function,
orthogonal to both {ψ 0 , ψ 1 }, collecting all higher than ψ 1 terms (the last terms
forming U) of Eqs. (13.1) and (13.2) for φ n = φ 0 . For φ 0 ,φ 1 very close to ψ 0 ,ψ 1 ,
as directly verified by expanding about φ 0 as well, the condition is satisfied when
ψ 0 | φ 1 2 ≤ ≤ψ 1 | φ 0 2 , i.e., when φ 1 attains better orthogonality to ψ 0 than φ 0 to
ψ 1 , which is essentially the original assumption that our φ 1 is more accurate than
our φ 0 .
In fact, if our φ 1 is close to ψ 1 , then a 2 × 2 diagonalization of the Hamiltonian
between φ 1 and φ 0 (or φ
+
0 ) should give as a higher eigenvector φ 1 ≈ ψ 1 unaffected
N. C. Bacalis
with energy
E
φ
+
0
= E [φ 0 ] −
(E [ψ 1 ] − E [φ 0 ]) ψ 1 |φ 0 2
1 − ψ 1 |φ 0 2
≤ E [φ 0 ]
(13.13)
(i.e., − is same or better than φ 0 ). Further, rotating φ
+
0 around ψ 1 would
improve φ
+
0 as follows: Introduce (e.g., by one more configuration) a function φ
(2+)
0
orthogonal to both {φ
+
0 , ψ 1 }. Then, in the subspace of {φ
+
0 , φ
(2+)
0
}, (both orthogonal
to ψ 1 ), the Hamiltonian opens their energy gap, so the lowest eigenvector − ≡ φ
−
0
has energy E
φ
−
0
≤ E
φ
+
0
, closer to E[ψ 0 ] (in a three-dimensional function
space {ψ 0 , ψ 1 , ψ k }, this would be exactly E[ψ 0 ] as directly verified). E
φ
−
0
could
be further improved by further rotating around ψ 1 similarly: i.e., after introducing
another function φ
(3+)
0
orthogonal to both {φ
−
0 , ψ 1 } in the subspace of {φ
−
0 , φ
(3+)
0
}
(both orthogonal to ψ 1 ), the lowest Hamiltonian eigenvector is − ≡ φ
(2−)
0
with
energy E
φ
(2−)
0
≤ E
φ
−
0
(even closer to E[ψ 0 ]) and so on.
Now let us consider the approximants.
13.4.2 Improving φ 0 Orthogonally to φ 1
Since ψ 1 is never exactly known, we can still improve φ 0 orthogonally to φ 1 , our
best approximant of ψ 1 (obtained either via F 1 or via a “large” HUM expansion),
by first computing φ
+
0 orthogonal to our φ 1 ,
φ
+
0 ≡
φ 0 − φ 1 φ 1 |φ 0
1 − φ 1 |φ 0 2
,
(13.14)
if the condition E
φ
+
0
≤ E [φ 0 ] is attainable: Indeed, by expanding about
φ 1 , as directly verified, the condition E
φ
+
0
≤ E [φ 0 ], to leading order, reads
(E 1 − E 0 )
1 − ψ 1 |φ 0 2 ≥
E
φ
⊥{1}
0
− E 0
φ
⊥{1}
0
|φ 0
2
, which is not impossible. Here, as defined in Eqs. (13.9) and (13.10), φ
⊥{1}
0
is the normalized function,
orthogonal to both {ψ 0 , ψ 1 }, collecting all higher than ψ 1 terms (the last terms
forming U) of Eqs. (13.1) and (13.2) for φ n = φ 0 . For φ 0 ,φ 1 very close to ψ 0 ,ψ 1 ,
as directly verified by expanding about φ 0 as well, the condition is satisfied when
ψ 0 | φ 1 2 ≤ ≤ψ 1 | φ 0 2 , i.e., when φ 1 attains better orthogonality to ψ 0 than φ 0 to
ψ 1 , which is essentially the original assumption that our φ 1 is more accurate than
our φ 0 .
In fact, if our φ 1 is close to ψ 1 , then a 2 × 2 diagonalization of the Hamiltonian
between φ 1 and φ 0 (or φ
+
0 ) should give as a higher eigenvector φ 1 ≈ ψ 1 unaffected