470
N. C. Bacalis
Why “unfortunately”? Because by optimizing one root of the secular equation,
all others get deteriorated: For example, the expansion of ψ 0 in the truncated basis
of the roots – all mutually orthogonal to each other – 0 , 1 , ..., N , leaves an
unknown remainder Y; therefore, even if we optimize at will one of them, say 1 ,
ψ 1 | 1 2 → 1, we shall have
ψ 0 | 0 2 + ψ 0 | 1 2 + ψ 0 | 2 2 + · · · + ψ 0 | N 2 ≤ 1 − Y 2
⇒ ψ 0 | 0 2 < 1 − ψ 0 | 2 2 − · · · − ψ 0 | N 2 < 1
i.e., 0 will be deteriorated, without being able to approach ψ 0 at will, and
the optimized 1 will be orthogonal to a deteriorated 0 . So, if the expansion
is large, then it tends to the exact eigenfunction, but if the expansion is small
(necessarily for large systems, like molecules, catalysts, etc.), then, in a truncated basis, it is worse than just always being “ 1 = ψ 1. ” For example, in a
subspace of mainly two configurations, if the deteriorated normalized function is
0 = αψ 0 + βψ 1 (+corrections), where β is not negligible, α <
1 − β 2 , then
the HUM-optimized (i.e., with E[ 1 ] > E 1 ) 1 = αψ 1 − βψ 0 (+corrections)
is equally deteriorated, and the worse, it is not even close to
+
1 , which is the
closest to the saddle point ψ 1 , among all normalized functions orthogonal to
0 :
+
1 =
1 − β 2
ψ 1 − αβψ 0
/
1 − β 2 (+corrections) with E
+
1
< E 1 ,
because the HUM theorem necessitates that the optimized root 1 shall not go below
E 1 (in order to try to approach
+
1 ).
Therefore, the HUM-optimized root 1 is much more away from the saddle point
ψ 1 , although its energy approaches E 1 . (Since there are normalized functions
orthogonal to φ 0 with E[] < E 1 , it will eventually lead to one of the infinitely
many normalized functions having E[] = E 1 . Indeed, doing the analysis from
any function if we consider a normalized function φ
⊥+
1
orthogonal to both φ 0
and φ 1
+ and diagonalize the Hamiltonian operator between φ
⊥+
1
and φ 1
+ , this will
open their energy gap, giving eigenfunctions − and + (both orthogonal to φ 0 ).
Then the function
=
−
E
+
− E 1
E
+
− E
−
±
+
E 1 − E
−
E
+
− E
−
is orthogonal to φ 0 and has equal to E 1 energy E[] = E 1 , without being ψ 1 . Even
worse, from any , consider a normalized function ⊥ orthogonal to both ψ 0 and
ψ 1 . Then the function
=
E [ ⊥ ] − E 1
E [ ⊥ ] − E 0
ψ 0 + 0ψ 1 −
E 1 − E 0
E [ ⊥ ] − E 0
⊥
has equal to E 1 energy: E[] = E 1 , but is orthogonal, to ψ 1 .) Thus, the only
rescue of HUM optimization of a higher root, in order to approach the excited state
N. C. Bacalis
Why “unfortunately”? Because by optimizing one root of the secular equation,
all others get deteriorated: For example, the expansion of ψ 0 in the truncated basis
of the roots – all mutually orthogonal to each other – 0 , 1 , ..., N , leaves an
unknown remainder Y; therefore, even if we optimize at will one of them, say 1 ,
ψ 1 | 1 2 → 1, we shall have
ψ 0 | 0 2 + ψ 0 | 1 2 + ψ 0 | 2 2 + · · · + ψ 0 | N 2 ≤ 1 − Y 2
⇒ ψ 0 | 0 2 < 1 − ψ 0 | 2 2 − · · · − ψ 0 | N 2 < 1
i.e., 0 will be deteriorated, without being able to approach ψ 0 at will, and
the optimized 1 will be orthogonal to a deteriorated 0 . So, if the expansion
is large, then it tends to the exact eigenfunction, but if the expansion is small
(necessarily for large systems, like molecules, catalysts, etc.), then, in a truncated basis, it is worse than just always being “ 1 = ψ 1. ” For example, in a
subspace of mainly two configurations, if the deteriorated normalized function is
0 = αψ 0 + βψ 1 (+corrections), where β is not negligible, α <
1 − β 2 , then
the HUM-optimized (i.e., with E[ 1 ] > E 1 ) 1 = αψ 1 − βψ 0 (+corrections)
is equally deteriorated, and the worse, it is not even close to
+
1 , which is the
closest to the saddle point ψ 1 , among all normalized functions orthogonal to
0 :
+
1 =
1 − β 2
ψ 1 − αβψ 0
/
1 − β 2 (+corrections) with E
+
1
< E 1 ,
because the HUM theorem necessitates that the optimized root 1 shall not go below
E 1 (in order to try to approach
+
1 ).
Therefore, the HUM-optimized root 1 is much more away from the saddle point
ψ 1 , although its energy approaches E 1 . (Since there are normalized functions
orthogonal to φ 0 with E[] < E 1 , it will eventually lead to one of the infinitely
many normalized functions having E[] = E 1 . Indeed, doing the analysis from
any function if we consider a normalized function φ
⊥+
1
orthogonal to both φ 0
and φ 1
+ and diagonalize the Hamiltonian operator between φ
⊥+
1
and φ 1
+ , this will
open their energy gap, giving eigenfunctions − and + (both orthogonal to φ 0 ).
Then the function
=
−
E
+
− E 1
E
+
− E
−
±
+
E 1 − E
−
E
+
− E
−
is orthogonal to φ 0 and has equal to E 1 energy E[] = E 1 , without being ψ 1 . Even
worse, from any , consider a normalized function ⊥ orthogonal to both ψ 0 and
ψ 1 . Then the function
=
E [ ⊥ ] − E 1
E [ ⊥ ] − E 0
ψ 0 + 0ψ 1 −
E 1 − E 0
E [ ⊥ ] − E 0
⊥
has equal to E 1 energy: E[] = E 1 , but is orthogonal, to ψ 1 .) Thus, the only
rescue of HUM optimization of a higher root, in order to approach the excited state