496
N. C. Bacalis
Obviously, fulfilling the inherent restrictions of the HUM functions [84], the 8term 2nd HUM root 1 (1,1,1,1,1) has lower quality than 0 (1,1,1,1,1) (≈ 0 (1,2,1,1,1) ) since
0.875 2 < (1 − (1.3 × 10 −5 ) 2 ), but also the optimized HUM 2nd root 1 (1,2,1,1,1)a
has lower quality as well, since 0.9986 2 < 1− (1.3 × 10 −5 ) 2 – assuming near
orthogonality (0.0026) to the optimized HUM 1st root.
“F 1 ”Root 8 Terms, a 1,2,1 = 0
In contrast, by minimizing F 1 , with the same set of parameters, (z 1,1 , z 1,2 , a 1,2,1 ),
a much better 8-term approximant 1 (1,2,1,1,1)F,a is obtained: (tSzaR = 111F)
E = −2.145152, F = −2.145151, F – E = 5.3 × 10 −7 , z 1,1 = 1.938718,
z 1,2 = 1.817736, a 1,2,1 = 0.799286, {c} = {1., 0.196506, 0.086316, −0.026204,
0.043226, 0.016366, 0.015810, 0.002479}, N = 1/3.637910, where the main
orbitals resemble 1s2s [cf. Figure 13.5]. The overlap with the optimized 8-term
HUM 1st root is 0 (1, 2, 1, 1, 1) | 1 (1, 2, 1, 1, 1)F, a = 0.0186 = (010H|111F). The
corresponding overlaps with the above-established basis are ψ 0 | 1 (1, 2, 1, 1, 1)F, a =
0.00490, and ψ 1 | 1 (1, 2, 1, 1, 1)F, a = 0.999807. Clearly, the F 1 -minimizing function,
although truncated (to 8 terms), is much closer to the exact saddle point. This
is also depicted in Fig. 13.7, where it satisfies the Schrödinger equation better
(−0.0003 < Schr < 0.0009) than the 8-term HUM 2nd root (0.0006 < Schr < 0.0012).
13.6.1.4 The Main Orbitals: F 1 (and OO) Give 1s2s; HUM Gives 1s1s
Observe that the HUM 2nd root’s main orbitals are 1s1s and it is “corrected” by
the c-series, the (OO) function, although veered away and below the saddle point
(thus, abandoned) has correct main orbitals 1s2s (cf. Eq. 13.18 ff), whereas F 1
directly finds physically correct main orbitals 1s2s (and better energy in truncated
space). If this HUM phenomenon emerges already in a system of two-electrons,
or three-electrons as foresaid and will be seen for Li (cf. Table 13.1), there is no
guarantee (and might be more important) in large systems, where small truncated
space is unavoidable and where the nature of the main (HOMO/LUMO) orbitals is
decisive in order to predict or cause desired reactions; then, the F functional will
be needed. The above energies and overlaps are summarized in Table 13.3.
13.6.1.5 Fulfillment of the Saddle Point Criteria by F 1 and Immediate
Improvement of 0
The functions obtained by F 1 , fulfill the saddle point criteria mentioned in the
introduction:
Firstly, to check the 27-term approximant ψ 1 , a 2 × 2 (H ij − ES ij ) generalized
diagonalization between ψ 1 and the 1-term 0 indeed leaves ψ 1 practically
unaffected, as “high 2 × 2 root” φ 1 = (ψ 1 –4.7 × 10 −5 0 ), with energy changed
by only 1.57 × 10 −9 a.u, while, by opening their energy “gap,” improves 0
N. C. Bacalis
Obviously, fulfilling the inherent restrictions of the HUM functions [84], the 8term 2nd HUM root 1 (1,1,1,1,1) has lower quality than 0 (1,1,1,1,1) (≈ 0 (1,2,1,1,1) ) since
0.875 2 < (1 − (1.3 × 10 −5 ) 2 ), but also the optimized HUM 2nd root 1 (1,2,1,1,1)a
has lower quality as well, since 0.9986 2 < 1− (1.3 × 10 −5 ) 2 – assuming near
orthogonality (0.0026) to the optimized HUM 1st root.
“F 1 ”Root 8 Terms, a 1,2,1 = 0
In contrast, by minimizing F 1 , with the same set of parameters, (z 1,1 , z 1,2 , a 1,2,1 ),
a much better 8-term approximant 1 (1,2,1,1,1)F,a is obtained: (tSzaR = 111F)
E = −2.145152, F = −2.145151, F – E = 5.3 × 10 −7 , z 1,1 = 1.938718,
z 1,2 = 1.817736, a 1,2,1 = 0.799286, {c} = {1., 0.196506, 0.086316, −0.026204,
0.043226, 0.016366, 0.015810, 0.002479}, N = 1/3.637910, where the main
orbitals resemble 1s2s [cf. Figure 13.5]. The overlap with the optimized 8-term
HUM 1st root is 0 (1, 2, 1, 1, 1) | 1 (1, 2, 1, 1, 1)F, a = 0.0186 = (010H|111F). The
corresponding overlaps with the above-established basis are ψ 0 | 1 (1, 2, 1, 1, 1)F, a =
0.00490, and ψ 1 | 1 (1, 2, 1, 1, 1)F, a = 0.999807. Clearly, the F 1 -minimizing function,
although truncated (to 8 terms), is much closer to the exact saddle point. This
is also depicted in Fig. 13.7, where it satisfies the Schrödinger equation better
(−0.0003 < Schr < 0.0009) than the 8-term HUM 2nd root (0.0006 < Schr < 0.0012).
13.6.1.4 The Main Orbitals: F 1 (and OO) Give 1s2s; HUM Gives 1s1s
Observe that the HUM 2nd root’s main orbitals are 1s1s and it is “corrected” by
the c-series, the (OO) function, although veered away and below the saddle point
(thus, abandoned) has correct main orbitals 1s2s (cf. Eq. 13.18 ff), whereas F 1
directly finds physically correct main orbitals 1s2s (and better energy in truncated
space). If this HUM phenomenon emerges already in a system of two-electrons,
or three-electrons as foresaid and will be seen for Li (cf. Table 13.1), there is no
guarantee (and might be more important) in large systems, where small truncated
space is unavoidable and where the nature of the main (HOMO/LUMO) orbitals is
decisive in order to predict or cause desired reactions; then, the F functional will
be needed. The above energies and overlaps are summarized in Table 13.3.
13.6.1.5 Fulfillment of the Saddle Point Criteria by F 1 and Immediate
Improvement of 0
The functions obtained by F 1 , fulfill the saddle point criteria mentioned in the
introduction:
Firstly, to check the 27-term approximant ψ 1 , a 2 × 2 (H ij − ES ij ) generalized
diagonalization between ψ 1 and the 1-term 0 indeed leaves ψ 1 practically
unaffected, as “high 2 × 2 root” φ 1 = (ψ 1 –4.7 × 10 −5 0 ), with energy changed
by only 1.57 × 10 −9 a.u, while, by opening their energy “gap,” improves 0