494
N. C. Bacalis
of the exponent z 0 . Indeed, 2r
1 has slightly better Schr values than F
1 , contrary to
the more truncated versions.
Thus, a quite reliable basis (27 terms) has been established, adequate for the
attempted demonstration, by adopting {ψ 0 = 0 (1,1,2,2,2) , ψ 1 = 1 (1,2,2,2,2)a = 0 }, since
the state-specific expansion, despite its more reasonable main orbitals, is slightly
inferior in Schr than the expansion of the (large) HUM 2nd root.
(n s , n t , n u ) = (1, 1, 1): 8 Terms
Minimal Basis
Ground State 8 Terms
The optimized ground state 8-term approximant 0 (1,1,1,1,1) has lowest root
energy (cf. Table 13.3: tSzaR = 0000) E[0 (1,1,1,1,1) ] = −2.903121 a.u. and
parameters z 0,1 = 1.84250, c 0,0,0,0 = 1, c 0,0,0,1 = 0.290798, c 0,0,1,0 = 0.190212,
c 0,0,1,1 = −0.0765151, c 0,1,0,0 = 0.00688611, c 0,1,0,1 = 0.0139948, c 0,1,1,0
= 0.016833, c 0,1,1,1 = 0.0120625, normalization constant N = 1/1.47861, while
its orthogonal 2nd root 1
0 (1,1,1,1,1)
(same z 0,1 ) lies too high: (tSzaR = 0001)
E[1]
0 (1,1,1,1,1)
= −2.01016 a.u. Since 0 (1,1,1,1,1) has ψ 0 | 0 (1, 1, 1, 1, 1) = 0.999958
and ψ 1 | 0 (1, 1, 1, 1, 1) = 1.3 × 10 −5 , it is nearly “perfect.” However, for the 1st
excited state, as seen below, this truncation (1,1,1,1,1) is not adequate: A richer
function (1,2,1,1,1) is needed.
2nd HUM Root and “F 1 ” Root 8 Terms
The optimized 8-term 2nd HUM root 1 (1,1,1,1,1) (see also Fig. 13.5) has
(tSzaR = 100H) E[1 (1,1,1,1,1) ] = −2.07215 a.u., z 1,1 = 1.44234, and
corresponding c-expansion coefficients {c} = {1, 0.199163, 0.0540545, 0.0273041,
−0.418026, −0.0402484, −0.0720681, −0.040584}, N = 1/1.3566, while
its orthogonal 1st root 0
1 (1,1,1,1,1)
(same z 1,1 ) is, of course, deteriorated:
(tSzaR = 1000) E[0]
1 (1,1,1,1,1)
= −2.89748 a.u., {c} = {1, 0.145873,
0.160241, −0.0416745, −0.245673, −0.00154526, −0.0197971, 0.00659241}.
The overlap between the normalized independently optimized approximants is
0 (1, 1, 1, 1, 1) | 1 (1, 1, 1, 1, 1) = − 1.97349 × 10 −5 = (ψ 0 |100H), meaning that the
HUM root 1 (1,1,1,1,1) has contributions from other higher states orthogonal to the
1st root (its energy is close to the 3rd). The overlap with the above established
ψ 1 = 1 (1,2,2,2,2) is ψ 1 | 1 (1, 1, 1, 1, 1) = 0.875. By using F 1 the optimized “F” root
1 (1,1,1,1,1)F (tSzaR = 100F), in lack of other parameters, nearly coincides with the
2nd HUM root near the minimum.
Richer Basis
In a richer parameter space, including z 1,2 and, optionally, a 1,2,1 = 0 (state-specific
description), the optimized 8-term functions are as follows:
N. C. Bacalis
of the exponent z 0 . Indeed, 2r
1 has slightly better Schr values than F
1 , contrary to
the more truncated versions.
Thus, a quite reliable basis (27 terms) has been established, adequate for the
attempted demonstration, by adopting {ψ 0 = 0 (1,1,2,2,2) , ψ 1 = 1 (1,2,2,2,2)a = 0 }, since
the state-specific expansion, despite its more reasonable main orbitals, is slightly
inferior in Schr than the expansion of the (large) HUM 2nd root.
(n s , n t , n u ) = (1, 1, 1): 8 Terms
Minimal Basis
Ground State 8 Terms
The optimized ground state 8-term approximant 0 (1,1,1,1,1) has lowest root
energy (cf. Table 13.3: tSzaR = 0000) E[0 (1,1,1,1,1) ] = −2.903121 a.u. and
parameters z 0,1 = 1.84250, c 0,0,0,0 = 1, c 0,0,0,1 = 0.290798, c 0,0,1,0 = 0.190212,
c 0,0,1,1 = −0.0765151, c 0,1,0,0 = 0.00688611, c 0,1,0,1 = 0.0139948, c 0,1,1,0
= 0.016833, c 0,1,1,1 = 0.0120625, normalization constant N = 1/1.47861, while
its orthogonal 2nd root 1
0 (1,1,1,1,1)
(same z 0,1 ) lies too high: (tSzaR = 0001)
E[1]
0 (1,1,1,1,1)
= −2.01016 a.u. Since 0 (1,1,1,1,1) has ψ 0 | 0 (1, 1, 1, 1, 1) = 0.999958
and ψ 1 | 0 (1, 1, 1, 1, 1) = 1.3 × 10 −5 , it is nearly “perfect.” However, for the 1st
excited state, as seen below, this truncation (1,1,1,1,1) is not adequate: A richer
function (1,2,1,1,1) is needed.
2nd HUM Root and “F 1 ” Root 8 Terms
The optimized 8-term 2nd HUM root 1 (1,1,1,1,1) (see also Fig. 13.5) has
(tSzaR = 100H) E[1 (1,1,1,1,1) ] = −2.07215 a.u., z 1,1 = 1.44234, and
corresponding c-expansion coefficients {c} = {1, 0.199163, 0.0540545, 0.0273041,
−0.418026, −0.0402484, −0.0720681, −0.040584}, N = 1/1.3566, while
its orthogonal 1st root 0
1 (1,1,1,1,1)
(same z 1,1 ) is, of course, deteriorated:
(tSzaR = 1000) E[0]
1 (1,1,1,1,1)
= −2.89748 a.u., {c} = {1, 0.145873,
0.160241, −0.0416745, −0.245673, −0.00154526, −0.0197971, 0.00659241}.
The overlap between the normalized independently optimized approximants is
0 (1, 1, 1, 1, 1) | 1 (1, 1, 1, 1, 1) = − 1.97349 × 10 −5 = (ψ 0 |100H), meaning that the
HUM root 1 (1,1,1,1,1) has contributions from other higher states orthogonal to the
1st root (its energy is close to the 3rd). The overlap with the above established
ψ 1 = 1 (1,2,2,2,2) is ψ 1 | 1 (1, 1, 1, 1, 1) = 0.875. By using F 1 the optimized “F” root
1 (1,1,1,1,1)F (tSzaR = 100F), in lack of other parameters, nearly coincides with the
2nd HUM root near the minimum.
Richer Basis
In a richer parameter space, including z 1,2 and, optionally, a 1,2,1 = 0 (state-specific
description), the optimized 8-term functions are as follows: