13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
495
a 1,2,1 = 0
HUM 1st Root 8 Terms, a 1,2,1 = 0
The optimized 8-term HUM 1st root 0 (1,2,1,1,1) has (tSzaR = 010H)
E = −2.903123 a.u, z 0,1 = 1.93393, z 0,2 = 3.50164, {c} = {1, 0.290154, 0.184912,
−0.077204, 0.006390, 0.014435, 0.017008, 0.011521}, N = 1/2.94444, with 2nd
root severely deteriorated (tSzaR = 0101), E = −2.0196 a.u. Thus, the 8-term
0 (1,2,1,1,1) is not essentially improved over the above simpler 8-term 0 (1,1,1,1,1) .
HUM 2nd Root 8 Terms, a 1,2,1 = 0
However, the optimized 8-term HUM 2nd root 1 (1,2,1,1,1) has (tSzaR = 110H)
E = −2.14449 a.u., z 1,1 = 0.851359, z 1,2 = 3.73405 (or z 1,1 = 1.86703,
z 1,2 = 1.70272), {c} = {1, 0.180637, −0.127452, 0.019460, −0.367640,
−0.065135, −0.046299, −0.003814}, N = 1/3.19489, with 1st root deteriorated
(tSzaR = 1100), E = −2.8886 a.u. The overlap 0 (1, 2, 1, 1, 1) | 1 (1, 2, 1, 1, 1) = 0.0026
=(010H|110H).
“F 1 ” Root 8 Terms, a 1,2,1 = 0
By minimizing F 1 (using the above fixed 1-term φ 0 ) first with a 1,2,1 = 0, the same
function is obtained, because the 1st root lies below φ 0 – and the lowest root above
E[φ 0 ] is the 2nd root. [With a 1,2,1 = 0 there are no other parameters to vary, and, as
seen in the next subsection, with a 1,2,1 = 0, a minimum of F 1 is obtained with the
same φ 0 above; so, it is not necessary to use any lower (fixed) φ 0 .]
a 1,2,1 = 0
HUM 2nd Root, 8 Terms, a 1,2,1 = 0
Using a 1,2,1 = 0, the optimized 8-term HUM 2nd root 1 (1,2,1,1,1)a , having
a 1,2,1 = 1.5773 × 10 −7 , remains essentially the same: (tSzaR = 111H)
E = −2.14449, z 1,1 = 1.86703, z 1,2 = 1.70272. The main orbitals resemble
1s1s (cf. Fig. 13.5). The overlaps with the above established ψ 0 = 0 (1,2,2,2,2)
and ψ 1 = 1 (1,2,2,2,2) are ψ 0 | 1 (1, 2, 1, 1, 1) = 0.0033, and ψ 1 | 1 (1, 2, 1, 1, 1) = 0.9986.
Generally, in a complete (infinite) space, both HUM and F should yield the same
wave function, but in a truncated space, the 2nd HUM root may be worse than
the lowest root above the known (highest) E[φ n−1 ]. Indeed, in 8-term He 2
1 S, at
the F 1 minimum, the lowest three roots above the known 1-term E[φ 0 ] = −2.824
are {−2.145, −2.028, −1.898}, whereas in optimizing the 2nd HUM root, the
lowest three HUM roots are {−2.889, −2.144, −1.993} (As seen later in detail,
F 1 yields E[φ 1 ] = −2.1452, much closer to the exact −2.1459 than the 2nd HUM
root E
φ 2r
1
= −2.144, while its 1st HUM root, E
φ 1r
1
= −2.889, is much
deteriorated).
495
a 1,2,1 = 0
HUM 1st Root 8 Terms, a 1,2,1 = 0
The optimized 8-term HUM 1st root 0 (1,2,1,1,1) has (tSzaR = 010H)
E = −2.903123 a.u, z 0,1 = 1.93393, z 0,2 = 3.50164, {c} = {1, 0.290154, 0.184912,
−0.077204, 0.006390, 0.014435, 0.017008, 0.011521}, N = 1/2.94444, with 2nd
root severely deteriorated (tSzaR = 0101), E = −2.0196 a.u. Thus, the 8-term
0 (1,2,1,1,1) is not essentially improved over the above simpler 8-term 0 (1,1,1,1,1) .
HUM 2nd Root 8 Terms, a 1,2,1 = 0
However, the optimized 8-term HUM 2nd root 1 (1,2,1,1,1) has (tSzaR = 110H)
E = −2.14449 a.u., z 1,1 = 0.851359, z 1,2 = 3.73405 (or z 1,1 = 1.86703,
z 1,2 = 1.70272), {c} = {1, 0.180637, −0.127452, 0.019460, −0.367640,
−0.065135, −0.046299, −0.003814}, N = 1/3.19489, with 1st root deteriorated
(tSzaR = 1100), E = −2.8886 a.u. The overlap 0 (1, 2, 1, 1, 1) | 1 (1, 2, 1, 1, 1) = 0.0026
=(010H|110H).
“F 1 ” Root 8 Terms, a 1,2,1 = 0
By minimizing F 1 (using the above fixed 1-term φ 0 ) first with a 1,2,1 = 0, the same
function is obtained, because the 1st root lies below φ 0 – and the lowest root above
E[φ 0 ] is the 2nd root. [With a 1,2,1 = 0 there are no other parameters to vary, and, as
seen in the next subsection, with a 1,2,1 = 0, a minimum of F 1 is obtained with the
same φ 0 above; so, it is not necessary to use any lower (fixed) φ 0 .]
a 1,2,1 = 0
HUM 2nd Root, 8 Terms, a 1,2,1 = 0
Using a 1,2,1 = 0, the optimized 8-term HUM 2nd root 1 (1,2,1,1,1)a , having
a 1,2,1 = 1.5773 × 10 −7 , remains essentially the same: (tSzaR = 111H)
E = −2.14449, z 1,1 = 1.86703, z 1,2 = 1.70272. The main orbitals resemble
1s1s (cf. Fig. 13.5). The overlaps with the above established ψ 0 = 0 (1,2,2,2,2)
and ψ 1 = 1 (1,2,2,2,2) are ψ 0 | 1 (1, 2, 1, 1, 1) = 0.0033, and ψ 1 | 1 (1, 2, 1, 1, 1) = 0.9986.
Generally, in a complete (infinite) space, both HUM and F should yield the same
wave function, but in a truncated space, the 2nd HUM root may be worse than
the lowest root above the known (highest) E[φ n−1 ]. Indeed, in 8-term He 2
1 S, at
the F 1 minimum, the lowest three roots above the known 1-term E[φ 0 ] = −2.824
are {−2.145, −2.028, −1.898}, whereas in optimizing the 2nd HUM root, the
lowest three HUM roots are {−2.889, −2.144, −1.993} (As seen later in detail,
F 1 yields E[φ 1 ] = −2.1452, much closer to the exact −2.1459 than the 2nd HUM
root E
φ 2r
1
= −2.144, while its 1st HUM root, E
φ 1r
1
= −2.889, is much
deteriorated).