13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
507
D a |
N
i>j
g i,j |D b =
1
√
D aa D bb
N
j>l
N
i>t
a i a t |g| b j b l
D ab
a i a t b j b l
where D ab (a i a t b j b l ) is the cofactor of D ab defined by deleting the rows and columns
containing a i | b j and a t | b l and attaching a factor (−1) i + j + t + l to the resultant
minor. The determinant cofactors are most efficiently computed via the inverse
matrix, after adding to all matrix elements small random numbers of the order
of machine accuracy, in order to avoid occasional (but harmless) vanishing of the
determinant.
13.6.3.1 Results
He
To check the reliability of the CI computation, before demonstrating comparisons
between F n and HUM – small and large – expansions, a comparison with the
literature is shown:
For the 1st excited state of He 1 S 1s2s, using, in φ 1 , 11 AVOLTOs forming 76
SDs and 22 configurations and a fixed crude φ 0 approximant of 1s 2 of only two
AVOLTOs in a 2 × 2 CI (E[φ 0 ] = −2.88 a.u.), then E[φ 1 ] = −2.1458140 a.u.
(F 1 = −2.1458139 a.u.). The wave function is primarily φ 1 = 0.9993 (1sB 2sA –
1sA 2sB) + 0.0190 (2p 1 A 3p −1 B – 2p 1 B 3p −1 A – 2p 0 A 3p 0 B + 2p 0 B 3p 0 A + 2p −1 A
3p 1 B – 2p −1 B 3p 1 A) + 0.0178 3s 2 , where A,B denote the spin, the {r rms distance
from the nucleus (a.u.); and z n, , b n, , q n, values} are, respectively, for 1s:
{0.8699666; 1.9690983, 0.0881525, 1.2060984}, for 2s: {5.6429592; 1.1248561,
0, 1}, for 2p: {1.1438142; 4.7885604, 0, 1}, and for 3s: {1.0879814; 5.6155136,
0, 1}, the g k -factors are for 2s: (0.8186816, 1) and for 3s: (4.0287651, 2.0322568,
1), making them both orthogonal to another 1s, a little more diffuse: {1.0374887;
3.0931600, −0.9341100, 0.9519696}. This energy value can be compared to (i)
−2.1457316 a.u., with 10 AVOLTOs forming 68 SDs and 18 configurations, exactly
orthogonal to a quite accurate φ 0 of E = −2.9031501 a.u. with 15 AVOLTOs, 157
SDs, and 40 configurations and to (ii) −2.145873 a.u. using 10 numerical MCHF
orbitals, with a comparably highly accurate ground state of E = −2.9031173 E h
[86].
For the 2nd excited state of He 1 S 1s3s, using, in φ 2 , 11 AVOLTOs, 77 SDs,
23 configurations, and again 2 × 2 φ 0 and φ 1 , then E[φ 2 ] = −2.0612263 a.u.
(F 1 = −2.0611758 a.u.). The wave function is primarily φ 2 = 0.9788 (1sA 3sB –
1sB 3sA) + 0.2035 (3sA 2sB – 3sB 2sA) + 0.0170 (1sA 2sB – 1sB 2sA); the {r rms ; z,
b, q} values are for 1s: {1.1051964; 1.61507, 2.0564, 0.95729}, for 3s: {12.9875477;
1.09458, 0, 1}, and for 2s: {1.6490337; 3.49456, 0, 1}; the g k -factors are for 3s:
(0.7660135, 0.9277702, 1), making it orthogonal to 1s and to the previous 2s, and for
2s: (1.5813926, 1), making it orthogonal to 1s. This energy value can be compared
to −2.0612681, obtained by B-splines [87]. By increasing to 19 AVOLTOs, 263
507
D a |
N
i>j
g i,j |D b =
1
√
D aa D bb
N
j>l
N
i>t
a i a t |g| b j b l
D ab
a i a t b j b l
where D ab (a i a t b j b l ) is the cofactor of D ab defined by deleting the rows and columns
containing a i | b j and a t | b l and attaching a factor (−1) i + j + t + l to the resultant
minor. The determinant cofactors are most efficiently computed via the inverse
matrix, after adding to all matrix elements small random numbers of the order
of machine accuracy, in order to avoid occasional (but harmless) vanishing of the
determinant.
13.6.3.1 Results
He
To check the reliability of the CI computation, before demonstrating comparisons
between F n and HUM – small and large – expansions, a comparison with the
literature is shown:
For the 1st excited state of He 1 S 1s2s, using, in φ 1 , 11 AVOLTOs forming 76
SDs and 22 configurations and a fixed crude φ 0 approximant of 1s 2 of only two
AVOLTOs in a 2 × 2 CI (E[φ 0 ] = −2.88 a.u.), then E[φ 1 ] = −2.1458140 a.u.
(F 1 = −2.1458139 a.u.). The wave function is primarily φ 1 = 0.9993 (1sB 2sA –
1sA 2sB) + 0.0190 (2p 1 A 3p −1 B – 2p 1 B 3p −1 A – 2p 0 A 3p 0 B + 2p 0 B 3p 0 A + 2p −1 A
3p 1 B – 2p −1 B 3p 1 A) + 0.0178 3s 2 , where A,B denote the spin, the {r rms distance
from the nucleus (a.u.); and z n, , b n, , q n, values} are, respectively, for 1s:
{0.8699666; 1.9690983, 0.0881525, 1.2060984}, for 2s: {5.6429592; 1.1248561,
0, 1}, for 2p: {1.1438142; 4.7885604, 0, 1}, and for 3s: {1.0879814; 5.6155136,
0, 1}, the g k -factors are for 2s: (0.8186816, 1) and for 3s: (4.0287651, 2.0322568,
1), making them both orthogonal to another 1s, a little more diffuse: {1.0374887;
3.0931600, −0.9341100, 0.9519696}. This energy value can be compared to (i)
−2.1457316 a.u., with 10 AVOLTOs forming 68 SDs and 18 configurations, exactly
orthogonal to a quite accurate φ 0 of E = −2.9031501 a.u. with 15 AVOLTOs, 157
SDs, and 40 configurations and to (ii) −2.145873 a.u. using 10 numerical MCHF
orbitals, with a comparably highly accurate ground state of E = −2.9031173 E h
[86].
For the 2nd excited state of He 1 S 1s3s, using, in φ 2 , 11 AVOLTOs, 77 SDs,
23 configurations, and again 2 × 2 φ 0 and φ 1 , then E[φ 2 ] = −2.0612263 a.u.
(F 1 = −2.0611758 a.u.). The wave function is primarily φ 2 = 0.9788 (1sA 3sB –
1sB 3sA) + 0.2035 (3sA 2sB – 3sB 2sA) + 0.0170 (1sA 2sB – 1sB 2sA); the {r rms ; z,
b, q} values are for 1s: {1.1051964; 1.61507, 2.0564, 0.95729}, for 3s: {12.9875477;
1.09458, 0, 1}, and for 2s: {1.6490337; 3.49456, 0, 1}; the g k -factors are for 3s:
(0.7660135, 0.9277702, 1), making it orthogonal to 1s and to the previous 2s, and for
2s: (1.5813926, 1), making it orthogonal to 1s. This energy value can be compared
to −2.0612681, obtained by B-splines [87]. By increasing to 19 AVOLTOs, 263