500
N. C. Bacalis
13.6.1.7 The Energy of φ
+
1
(the Closest to ψ 1 Orthogonal to φ 0 )
Now, by using the above-established 2r
1 as ψ 1 and a truncated approximant of ψ 0
as φ 0 , it is demonstrated that the corresponding
φ
+
1
[cf. Eq. 13.3], i.e., the closest
to ψ 1 while orthogonal to φ 0 , lies lower than E[ψ 1 ] (actually, than E[ 2r
1 ]). Since
the 8-term HUM 1st root 0 (1,1,1,1,1) is, as seen above, “almost perfect,” φ 0 | 1 2 =
10 −10 , E
φ
+
1
is lower than E 1 = −2.14584 by only 10 −10 a.u., well beyond
the accuracy of the present demonstration, so, a less accurate φ 0 will be used in
the demonstration, e.g., the 1-term “fixed,” used in F 1 , with φ 0 | 1= 0.04786;
then by replacing ψ 1 by 2r
1 in
φ
+
1
:
φ
+
1
H
φ
+
1
= −2.15470 a.u., whereas by
replacing it in the exact formula [cf. Eq. 13.3]: E
φ
+
1
= −2.14745 a.u. The 0.3%
discrepancy is due to the inexactness of 2r
1 . We expect that if the energy were
further minimized while keeping orthogonality to this 1-term φ 0 = 0 (0,1,0,0,0) , we
should end up with a function orthogonal to this φ 0 , lying lower than −2.15470 a.u.
(if this were the correct value based on the exact ψ 1 ), veered well away from ψ 1 .
This is demonstrated below. If the deteriorated HUM 1st root, orthogonal to the
optimized HUM 2nd root 1 (12111)a = 0 , is used as φ 0 , with φ 0 | 1 = −0.00405,
then the corresponding
φ
+
1
H
φ
+
1
= −2.14591 a.u. (and by the exact Eq.
13.3: E
φ
+
1
= − 2.14585 a.u.), which is slightly below E[ 2r
1 ], while the exact
eigenvalue is slightly even lower, which confirms that it is not necessary for the 1st
root to approach ψ 0 , φ 0 → ψ 0 , although deteriorated, only orthogonality to ψ 1 ,
φ 0 | ψ 1 → 0, is adequate.
13.6.1.8 Comparison Between F, HUM, and the Lowest Orthogonal to φ 0
(OO)
Up to now it has been demonstrated that the optimized HUM 2nd root 2r
1 is
veered away from the exact saddle point because it must be orthogonal to a
deteriorated approximant of the ground state while lying higher than the exact (in
accordance with the HUM theorem). Also, previously it was demonstrated that (OO)
minimizing the excited state energy orthogonally to a normalized approximant φ 0 ,
of ψ 0 , leads to a function, φ LE
1 , lying lower than the exact (not collapsed), therefore,
also veered away from the exact. Thus the F 1 -minimizing function, φ
F 1
1 , is the most
reliable truncated approximant among the three {φ
F 1
1 , φ LE
1 , and 2r
1 }.
13.6.1.9 Remarks
1. If 1| φ 0 = 0, then the three functions {φ
F 1
1 , φ LE
1 , and 2r
1 } coincide with ψ 1
and φ
+
1 , because the orthogonal to φ 0 subspace will contain ψ 1 ; hence φ
+
1
will coincide with ψ 1 : not lying lower. Thus the minimizing function φ LE
1 will
coincide with φ
+
1 and, therefore, with ψ 1 . Also, 2r
1 , having φ 0 as 1st root, will
N. C. Bacalis
13.6.1.7 The Energy of φ
+
1
(the Closest to ψ 1 Orthogonal to φ 0 )
Now, by using the above-established 2r
1 as ψ 1 and a truncated approximant of ψ 0
as φ 0 , it is demonstrated that the corresponding
φ
+
1
[cf. Eq. 13.3], i.e., the closest
to ψ 1 while orthogonal to φ 0 , lies lower than E[ψ 1 ] (actually, than E[ 2r
1 ]). Since
the 8-term HUM 1st root 0 (1,1,1,1,1) is, as seen above, “almost perfect,” φ 0 | 1 2 =
10 −10 , E
φ
+
1
is lower than E 1 = −2.14584 by only 10 −10 a.u., well beyond
the accuracy of the present demonstration, so, a less accurate φ 0 will be used in
the demonstration, e.g., the 1-term “fixed,” used in F 1 , with φ 0 | 1= 0.04786;
then by replacing ψ 1 by 2r
1 in
φ
+
1
:
φ
+
1
H
φ
+
1
= −2.15470 a.u., whereas by
replacing it in the exact formula [cf. Eq. 13.3]: E
φ
+
1
= −2.14745 a.u. The 0.3%
discrepancy is due to the inexactness of 2r
1 . We expect that if the energy were
further minimized while keeping orthogonality to this 1-term φ 0 = 0 (0,1,0,0,0) , we
should end up with a function orthogonal to this φ 0 , lying lower than −2.15470 a.u.
(if this were the correct value based on the exact ψ 1 ), veered well away from ψ 1 .
This is demonstrated below. If the deteriorated HUM 1st root, orthogonal to the
optimized HUM 2nd root 1 (12111)a = 0 , is used as φ 0 , with φ 0 | 1 = −0.00405,
then the corresponding
φ
+
1
H
φ
+
1
= −2.14591 a.u. (and by the exact Eq.
13.3: E
φ
+
1
= − 2.14585 a.u.), which is slightly below E[ 2r
1 ], while the exact
eigenvalue is slightly even lower, which confirms that it is not necessary for the 1st
root to approach ψ 0 , φ 0 → ψ 0 , although deteriorated, only orthogonality to ψ 1 ,
φ 0 | ψ 1 → 0, is adequate.
13.6.1.8 Comparison Between F, HUM, and the Lowest Orthogonal to φ 0
(OO)
Up to now it has been demonstrated that the optimized HUM 2nd root 2r
1 is
veered away from the exact saddle point because it must be orthogonal to a
deteriorated approximant of the ground state while lying higher than the exact (in
accordance with the HUM theorem). Also, previously it was demonstrated that (OO)
minimizing the excited state energy orthogonally to a normalized approximant φ 0 ,
of ψ 0 , leads to a function, φ LE
1 , lying lower than the exact (not collapsed), therefore,
also veered away from the exact. Thus the F 1 -minimizing function, φ
F 1
1 , is the most
reliable truncated approximant among the three {φ
F 1
1 , φ LE
1 , and 2r
1 }.
13.6.1.9 Remarks
1. If 1| φ 0 = 0, then the three functions {φ
F 1
1 , φ LE
1 , and 2r
1 } coincide with ψ 1
and φ
+
1 , because the orthogonal to φ 0 subspace will contain ψ 1 ; hence φ
+
1
will coincide with ψ 1 : not lying lower. Thus the minimizing function φ LE
1 will
coincide with φ
+
1 and, therefore, with ψ 1 . Also, 2r
1 , having φ 0 as 1st root, will