13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
471
saddle point, is to use huge expansions (α → 1) (impracticable for large systems).
The above are explained pictorially in Fig. 13.2.
However, in order to correctly comprehend charge transfer that will enable us
to guide the applied research toward the correct and not (as seen below) toward
a deceiving direction, we must find the saddle point, ψ n , and, as explained above,
with truncated bases, this cannot be approached by neither orthogonal optimization
(OO) nor by HUM optimization. On the contrary, by both of these methods
(OO, HUM), with truncated bases, the saddle point ψ n , is avoided! (At least,
by HUM the correct energy is approached). For this reason, a safe approach
of the saddle point, ψ n , is needed, using truncated bases, thus, allowing both
an understanding and a correct prediction, of practical interest, about the
behavior of the electrons concerning charge transfer. Specifically:
Because, by HUM, the space spanned by a small basis is not completely
exploitable, since, necessarily, normalized functions that are closer to the saddle
point ψ 1 – and in general to ψ n (i.e., with lower energy than the HUM answer,
toward +
n ) – are excluded (forbidden) by HUM, we shall see a developed
functional, F n [2], which has local minimum at the saddle points, ψ n , which uses
lower-lying normalized approximants i , i < n, but, practically, does not depend
on i ’s accuracy (provided only that the i s are reasonable):
F n [ 0 , , 1 , . . . ; n ] ≡E [ n ] +2
i
E [ n ] −E [ i ]
1−
i
2
−1
.
(13.4)
As shown below, F n has a minimum at each excited energy saddle point, even
in pathological cases where at the excited eigenvalue there is no definite Hessian
(to count its negative eigenvalues). As a “difficult” check, F n easily passes the
pathological Rellich test [5] of the Hermitian matrix:
− sin x sin y
sin y sin x
X
Y
= E
X
Y
⇒ E 0,1 = ±
sin
2 x + sin
2 y.
The “excited” E 1 has minimum at x = 0, y = 0 (as parameters), with indefinite
Hessian there. The eigenvectors ψ 0, ψ 1 are easily computed:
Y 0,1
X 0,1
=
±
sin
2 x + sin
2 y + sin x
sin y
, ψ 0,1 (x, y) =
X 0,1
Y 0,1
/
X 2
0,1 + Y 2
0,1 .
If we choose in F 1 (i.e., for n = 1), as fixed function, a crude but reasonable
“ground state” expansion, in terms of {ψ 0 , ψ 1 }, namely, 0 = 0.1 ψ 1 +
ψ 0
1 − 0.1
2 (where 0.1 is “close” to 0, but, obviously, not too close) and if,
by using this fixed function 0 , we minimize F 1 by optimizing 1 = a ψ 1 +
471
saddle point, is to use huge expansions (α → 1) (impracticable for large systems).
The above are explained pictorially in Fig. 13.2.
However, in order to correctly comprehend charge transfer that will enable us
to guide the applied research toward the correct and not (as seen below) toward
a deceiving direction, we must find the saddle point, ψ n , and, as explained above,
with truncated bases, this cannot be approached by neither orthogonal optimization
(OO) nor by HUM optimization. On the contrary, by both of these methods
(OO, HUM), with truncated bases, the saddle point ψ n , is avoided! (At least,
by HUM the correct energy is approached). For this reason, a safe approach
of the saddle point, ψ n , is needed, using truncated bases, thus, allowing both
an understanding and a correct prediction, of practical interest, about the
behavior of the electrons concerning charge transfer. Specifically:
Because, by HUM, the space spanned by a small basis is not completely
exploitable, since, necessarily, normalized functions that are closer to the saddle
point ψ 1 – and in general to ψ n (i.e., with lower energy than the HUM answer,
toward +
n ) – are excluded (forbidden) by HUM, we shall see a developed
functional, F n [2], which has local minimum at the saddle points, ψ n , which uses
lower-lying normalized approximants i , i < n, but, practically, does not depend
on i ’s accuracy (provided only that the i s are reasonable):
F n [ 0 , , 1 , . . . ; n ] ≡E [ n ] +2
i
E [ n ] −E [ i ]
1−
i
2
−1
.
(13.4)
As shown below, F n has a minimum at each excited energy saddle point, even
in pathological cases where at the excited eigenvalue there is no definite Hessian
(to count its negative eigenvalues). As a “difficult” check, F n easily passes the
pathological Rellich test [5] of the Hermitian matrix:
− sin x sin y
sin y sin x
X
Y
= E
X
Y
⇒ E 0,1 = ±
sin
2 x + sin
2 y.
The “excited” E 1 has minimum at x = 0, y = 0 (as parameters), with indefinite
Hessian there. The eigenvectors ψ 0, ψ 1 are easily computed:
Y 0,1
X 0,1
=
±
sin
2 x + sin
2 y + sin x
sin y
, ψ 0,1 (x, y) =
X 0,1
Y 0,1
/
X 2
0,1 + Y 2
0,1 .
If we choose in F 1 (i.e., for n = 1), as fixed function, a crude but reasonable
“ground state” expansion, in terms of {ψ 0 , ψ 1 }, namely, 0 = 0.1 ψ 1 +
ψ 0
1 − 0.1
2 (where 0.1 is “close” to 0, but, obviously, not too close) and if,
by using this fixed function 0 , we minimize F 1 by optimizing 1 = a ψ 1 +