Singularities of Møller-Plesset series: example "a30"
Molecule BH. Basis aug-cc-pVTZ. Structure "mpn_Rfci"
Content
- Definition of quadratic approximants
- Approximant [6, 6, 5]
- Approximant [6, 6, 6]
- Approximant [7, 6, 6]
- Approximant [7, 7, 6]
- Approximant [7, 7, 7]
- Approximant [8, 7, 7]
- Approximant [8, 8, 7]
- Approximant [8, 8, 8]
- Approximant [9, 8, 8]
- Approximant [9, 9, 8]
- Approximant [9, 9, 9]
- Approximant [10, 9, 9]
- Approximant [10, 10, 9]
- Approximant [10, 10, 10]
- Approximant [11, 10, 10]
- Approximant [11, 11, 10]
- Approximant [11, 11, 11]
- Approximant [12, 11, 11]
- Approximant [12, 12, 11]
- Approximant [12, 12, 12]
Examples | a1 | a2 | a8 | a16 | a22 | a30 | a38 | a44 | a45 | a51 | a62 | a69 | a75 | a83 | a84 | a85 | a86 | a87 | a88 | a90 | a91 |
Molecule | Ar | BH | BH | BH | BH | BH | BH | BO+ | C2 | CN+ | N2 | HF | HF | HCl | HCl | F- | Cl- | Cl- | Ne | OH- | SH- |
Basis | aug-cc-pVDZ | cc-pVDZ | cc-pVTZ | cc-pVQZ | aug-cc-pVDZ | aug-cc-pVTZ | aug-cc-pVQZ | cc-pVDZ | cc-pVDZ | cc-pVDZ | cc-pVDZ | cc-pVDZ | aug-cc-pVDZ | cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ | cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ |
Quadratic approximants
[n1, n2, n3] approximant is defined
as a solution of the quadratic equation
A(z)f2 + B(z)f + C(z) = 0
with polynomial coefficients A(z),
B(z) and
C(z) of degree
n3, n2 and n1 respectively.
Square-root singularities are determined as zeroes of the discriminant
D(z) = B2(z) - 4A(z)C(z).
The weight c of the singularity zc is defined so that
f ~ c(1 - z/zc)1/2 at z -> zc.
The weight is calculated by formula
c = 1/2[-z(D/A2)']1/2
where r. h. s. of the above equation is evaluated at z = zc.
Table 1. Singularities with their weights for the quadratic approximant [6, 6, 5] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 1.4297 + 0.022 i | 0.101 - 0.137 i |
![Singularities of quadratic [6, 6, 5] approximant](singsq1.gif?39954) |
2 | 1.4297 - 0.022 i | 0.101 + 0.137 i |
3 | 1.6579 + 0.4269 i | 0.176 + 0.157 i |
4 | 1.6579 - 0.4269 i | 0.176 - 0.157 i |
5 | 2.4747 | 8.67 |
6 | -2.9675 | 0.00524 |
7 | -2.7124 + 1.6161 i | 0.000715 + 0.0049 i |
8 | -2.7124 - 1.6161 i | 0.000715 - 0.0049 i |
9 | -2.7035 + 2.1193 i | 0.00663 - 0.00088 i |
10 | -2.7035 - 2.1193 i | 0.00663 + 0.00088 i |
11 | 18.0867 | 4.45 i |
12 | -57.7998 | 1.46 i |
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Table 2. Singularities with their weights for the quadratic approximant [6, 6, 6] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.2544 + 0.e-5 i | 6.27e-8 - 6.27e-8 i |
![Singularities of quadratic [6, 6, 6] approximant](singsq2.gif?485242) |
2 | 0.2544 - 0.e-5 i | 6.27e-8 + 6.27e-8 i |
3 | 1.4488 | 0.139 |
4 | 1.5142 | 0.765 i |
5 | 1.6549 + 0.4684 i | 0.0684 + 0.111 i |
6 | 1.6549 - 0.4684 i | 0.0684 - 0.111 i |
7 | 2.5653 | 5.24e4 |
8 | -2.8514 + 0.814 i | 0.004 + 0.00279 i |
9 | -2.8514 - 0.814 i | 0.004 - 0.00279 i |
10 | -3.2502 + 1.5582 i | 0.00425 - 0.0069 i |
11 | -3.2502 - 1.5582 i | 0.00425 + 0.0069 i |
12 | -6.5728 | 0.235 |
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Table 3. Singularities with their weights for the quadratic approximant [7, 6, 6] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.4278 + 0.e-5 i | 3.68e-7 + 3.68e-7 i |
![Singularities of quadratic [7, 6, 6] approximant](singsq3.gif?156605) |
2 | -0.4278 - 0.e-5 i | 3.68e-7 - 3.68e-7 i |
3 | 1.3852 | 0.0349 |
4 | 1.4994 | 0.0931 i |
5 | 1.5775 + 0.4526 i | 0.0601 + 0.0246 i |
6 | 1.5775 - 0.4526 i | 0.0601 - 0.0246 i |
7 | -3.0933 | 0.0158 |
8 | -2.2854 + 2.8671 i | 0.00722 - 0.00361 i |
9 | -2.2854 - 2.8671 i | 0.00722 + 0.00361 i |
10 | 3.9442 + 1.1436 i | 0.183 - 0.199 i |
11 | 3.9442 - 1.1436 i | 0.183 + 0.199 i |
12 | -3.7911 + 2.7039 i | 0.000483 + 0.00862 i |
13 | -3.7911 - 2.7039 i | 0.000483 - 0.00862 i |
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Table 4. Singularities with their weights for the quadratic approximant [7, 7, 6] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.2539 + 0.4889 i | 2.44e-7 + 4.e-8 i |
![Singularities of quadratic [7, 7, 6] approximant](singsq4.gif?991778) |
2 | -0.2539 - 0.4889 i | 2.44e-7 - 4.e-8 i |
3 | -0.2539 + 0.4889 i | 4.e-8 - 2.44e-7 i |
4 | -0.2539 - 0.4889 i | 4.e-8 + 2.44e-7 i |
5 | 1.0336 + 0.0022 i | 0.000139 - 0.000137 i |
6 | 1.0336 - 0.0022 i | 0.000139 + 0.000137 i |
7 | 1.4468 | 0.0193 |
8 | 1.8305 + 0.697 i | 0.0404 + 0.0382 i |
9 | 1.8305 - 0.697 i | 0.0404 - 0.0382 i |
10 | -2.3396 + 0.6743 i | 0.000511 + 0.000222 i |
11 | -2.3396 - 0.6743 i | 0.000511 - 0.000222 i |
12 | -3.6688 | 0.00131 |
13 | -6.5053 | 0.0178 i |
14 | 14.9597 | 4.58 i |
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Table 5. Singularities with their weights for the quadratic approximant [7, 7, 7] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.6159 + 0.e-5 i | 2.25e-7 + 2.25e-7 i |
![Singularities of quadratic [7, 7, 7] approximant](singsq5.gif?749878) |
2 | -0.6159 - 0.e-5 i | 2.25e-7 - 2.25e-7 i |
3 | 1.4167 | 0.0233 |
4 | -0.1364 + 1.5203 i | 0.0000567 - 0.0000396 i |
5 | -0.1364 - 1.5203 i | 0.0000567 + 0.0000396 i |
6 | -0.1372 + 1.5298 i | 0.0000395 + 0.0000577 i |
7 | -0.1372 - 1.5298 i | 0.0000395 - 0.0000577 i |
8 | 1.5602 + 0.634 i | 0.006 - 0.0148 i |
9 | 1.5602 - 0.634 i | 0.006 + 0.0148 i |
10 | 2.2502 | 0.168 i |
11 | -2.4368 + 0.7254 i | 0.000599 + 0.0000455 i |
12 | -2.4368 - 0.7254 i | 0.000599 - 0.0000455 i |
13 | -4.5002 | 0.00475 |
14 | 9.9755 | 1.57 |
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Table 6. Singularities with their weights for the quadratic approximant [8, 7, 7] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.6273 + 0.e-5 i | 2.86e-7 + 2.86e-7 i |
![Singularities of quadratic [8, 7, 7] approximant](singsq6.gif?73931) |
2 | -0.6273 - 0.e-5 i | 2.86e-7 - 2.86e-7 i |
3 | 1.4176 | 0.0237 |
4 | -0.045 + 1.5312 i | 0.0000464 - 0.0000671 i |
5 | -0.045 - 1.5312 i | 0.0000464 + 0.0000671 i |
6 | -0.0417 + 1.5399 i | 0.0000679 + 0.0000472 i |
7 | -0.0417 - 1.5399 i | 0.0000679 - 0.0000472 i |
8 | 1.5612 + 0.6323 i | 0.00582 - 0.0152 i |
9 | 1.5612 - 0.6323 i | 0.00582 + 0.0152 i |
10 | 2.2205 | 0.173 i |
11 | -2.5566 + 0.7565 i | 0.000884 - 0.0000102 i |
12 | -2.5566 - 0.7565 i | 0.000884 + 0.0000102 i |
13 | 7.1154 | 71.4 |
14 | -8.011 + 2.832 i | 0.0584 + 0.0762 i |
15 | -8.011 - 2.832 i | 0.0584 - 0.0762 i |
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Table 7. Singularities with their weights for the quadratic approximant [8, 8, 7] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.1674 | 0 |
![Singularities of quadratic [8, 8, 7] approximant](singsq7.gif?776843) |
2 | 0.1674 | 0 |
3 | -0.6678 | 6.3e-7 |
4 | -0.6678 | 6.3e-7 i |
5 | 1.4139 | 0.0188 |
6 | 1.5706 + 0.731 i | 0.0103 - 0.00376 i |
7 | 1.5706 - 0.731 i | 0.0103 + 0.00376 i |
8 | 0.8027 + 1.8393 i | 0.000814 - 0.0000936 i |
9 | 0.8027 - 1.8393 i | 0.000814 + 0.0000936 i |
10 | 0.903 + 1.9347 i | 0.0000916 + 0.00099 i |
11 | 0.903 - 1.9347 i | 0.0000916 - 0.00099 i |
12 | -2.8947 | 0.00439 |
13 | -2.6815 + 3.7051 i | 0.0134 + 0.00405 i |
14 | -2.6815 - 3.7051 i | 0.0134 - 0.00405 i |
15 | -4.9868 | 0.0219 i |
16 | 11.6828 | 0.398 i |
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Table 8. Singularities with their weights for the quadratic approximant [8, 8, 8] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.1085 + 0.6476 i | 1.09e-7 - 2.15e-7 i |
![Singularities of quadratic [8, 8, 8] approximant](singsq8.gif?647578) |
2 | -0.1085 - 0.6476 i | 1.09e-7 + 2.15e-7 i |
3 | -0.1085 + 0.6476 i | 2.15e-7 + 1.09e-7 i |
4 | -0.1085 - 0.6476 i | 2.15e-7 - 1.09e-7 i |
5 | -1.0572 + 0.4279 i | 6.99e-7 - 1.16e-6 i |
6 | -1.0572 - 0.4279 i | 6.99e-7 + 1.16e-6 i |
7 | -1.0629 + 0.4191 i | 1.14e-6 + 6.98e-7 i |
8 | -1.0629 - 0.4191 i | 1.14e-6 - 6.98e-7 i |
9 | 1.4214 | 0.0316 |
10 | 1.5569 + 0.5709 i | 0.00504 + 0.0208 i |
11 | 1.5569 - 0.5709 i | 0.00504 - 0.0208 i |
12 | 1.8458 | 1.34 i |
13 | -1.7293 + 0.6455 i | 0.0000238 - 0.0000129 i |
14 | -1.7293 - 0.6455 i | 0.0000238 + 0.0000129 i |
15 | -2.036 | 0.0000297 |
16 | 4.9725 | 0.758 |
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Table 9. Singularities with their weights for the quadratic approximant [9, 8, 8] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.1036 + 0.69 i | 1.62e-7 - 3.e-7 i |
![Singularities of quadratic [9, 8, 8] approximant](singsq9.gif?571669) |
2 | -0.1036 - 0.69 i | 1.62e-7 + 3.e-7 i |
3 | -0.1036 + 0.6901 i | 3.e-7 + 1.62e-7 i |
4 | -0.1036 - 0.6901 i | 3.e-7 - 1.62e-7 i |
5 | -0.8547 + 0.427 i | 2.49e-8 - 6.63e-7 i |
6 | -0.8547 - 0.427 i | 2.49e-8 + 6.63e-7 i |
7 | -0.8558 + 0.4266 i | 6.63e-7 + 2.62e-8 i |
8 | -0.8558 - 0.4266 i | 6.63e-7 - 2.62e-8 i |
9 | 1.4232 | 0.0343 |
10 | 1.5581 + 0.5622 i | 0.00728 + 0.0214 i |
11 | 1.5581 - 0.5622 i | 0.00728 - 0.0214 i |
12 | 1.8046 | 3.66 i |
13 | -1.9453 + 0.728 i | 0.0000737 - 0.0000352 i |
14 | -1.9453 - 0.728 i | 0.0000737 + 0.0000352 i |
15 | -2.7579 | 0.000227 |
16 | 4.5572 | 0.626 |
17 | -46.3729 | 0.179 i |
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Table 10. Singularities with their weights for the quadratic approximant [9, 9, 8] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.0572 + 0.6864 i | 1.94e-7 - 5.62e-9 i |
![Singularities of quadratic [9, 9, 8] approximant](singsq10.gif?482846) |
2 | -0.0572 - 0.6864 i | 1.94e-7 + 5.62e-9 i |
3 | -0.0572 + 0.6864 i | 5.61e-9 + 1.94e-7 i |
4 | -0.0572 - 0.6864 i | 5.61e-9 - 1.94e-7 i |
5 | -0.7467 + 0.3531 i | 1.26e-8 - 2.01e-7 i |
6 | -0.7467 - 0.3531 i | 1.26e-8 + 2.01e-7 i |
7 | -0.7469 + 0.3529 i | 2.01e-7 + 1.27e-8 i |
8 | -0.7469 - 0.3529 i | 2.01e-7 - 1.27e-8 i |
9 | 1.4405 | 0.0549 |
10 | 1.5293 + 0.5604 i | 0.00907 + 0.013 i |
11 | 1.5293 - 0.5604 i | 0.00907 - 0.013 i |
12 | 1.8201 | 1.58 i |
13 | -2.1835 + 0.9826 i | 0.0000748 - 0.000178 i |
14 | -2.1835 - 0.9826 i | 0.0000748 + 0.000178 i |
15 | -3.0645 + 4.0175 i | 0.000215 + 0.00444 i |
16 | -3.0645 - 4.0175 i | 0.000215 - 0.00444 i |
17 | 4.4049 + 5.3288 i | 0.0593 + 0.00741 i |
18 | 4.4049 - 5.3288 i | 0.0593 - 0.00741 i |
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Table 11. Singularities with their weights for the quadratic approximant [9, 9, 9] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.2904 | 4.63e-10 |
![Singularities of quadratic [9, 9, 9] approximant](singsq11.gif?142085) |
2 | 0.2904 | 4.63e-10 i |
3 | -0.0998 + 0.7068 i | 3.89e-8 + 2.42e-7 i |
4 | -0.0998 - 0.7068 i | 3.89e-8 - 2.42e-7 i |
5 | -0.0998 + 0.7068 i | 2.42e-7 - 3.89e-8 i |
6 | -0.0998 - 0.7068 i | 2.42e-7 + 3.89e-8 i |
7 | -0.8206 + 0.4201 i | 7.34e-8 + 3.7e-7 i |
8 | -0.8206 - 0.4201 i | 7.34e-8 - 3.7e-7 i |
9 | -0.8216 + 0.4197 i | 3.7e-7 - 7.26e-8 i |
10 | -0.8216 - 0.4197 i | 3.7e-7 + 7.26e-8 i |
11 | 1.4077 | 0.0221 |
12 | 1.5798 + 0.5686 i | 0.00113 + 0.0296 i |
13 | 1.5798 - 0.5686 i | 0.00113 - 0.0296 i |
14 | 1.8407 | 1.93 i |
15 | -1.9665 + 0.7729 i | 0.0000668 - 0.000053 i |
16 | -1.9665 - 0.7729 i | 0.0000668 + 0.000053 i |
17 | -2.967 | 0.000344 |
18 | 4.3754 | 0.541 |
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Table 12. Singularities with their weights for the quadratic approximant [10, 9, 9] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.0505 + 0.9211 i | 1.27e-7 + 7.6e-8 i |
![Singularities of quadratic [10, 9, 9] approximant](singsq12.gif?6253) |
2 | -0.0505 - 0.9211 i | 1.27e-7 - 7.6e-8 i |
3 | -0.0511 + 0.9234 i | 7.77e-8 - 1.27e-7 i |
4 | -0.0511 - 0.9234 i | 7.77e-8 + 1.27e-7 i |
5 | -0.4985 + 0.8071 i | 3.47e-8 - 6.74e-8 i |
6 | -0.4985 - 0.8071 i | 3.47e-8 + 6.74e-8 i |
7 | -0.5035 + 0.8064 i | 6.72e-8 + 3.53e-8 i |
8 | -0.5035 - 0.8064 i | 6.72e-8 - 3.53e-8 i |
9 | -1.2513 + 0.248 i | 2.08e-7 + 3.17e-7 i |
10 | -1.2513 - 0.248 i | 2.08e-7 - 3.17e-7 i |
11 | 1.3084 + 0.2478 i | 0.0000347 + 0.00106 i |
12 | 1.3084 - 0.2478 i | 0.0000347 - 0.00106 i |
13 | 1.3379 + 0.3934 i | 0.000353 + 0.00113 i |
14 | 1.3379 - 0.3934 i | 0.000353 - 0.00113 i |
15 | -1.4759 + 0.0205 i | 1.48e-7 + 1.23e-7 i |
16 | -1.4759 - 0.0205 i | 1.48e-7 - 1.23e-7 i |
17 | -1.145 + 1.8434 i | 0.0000206 - 0.0000143 i |
18 | -1.145 - 1.8434 i | 0.0000206 + 0.0000143 i |
19 | 11.9989 | 0.156 |
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Table 13. Singularities with their weights for the quadratic approximant [10, 10, 9] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.4539 + 0.5762 i | 1.94e-9 - 7.08e-9 i |
![Singularities of quadratic [10, 10, 9] approximant](singsq13.gif?152146) |
2 | -0.4539 - 0.5762 i | 1.94e-9 + 7.08e-9 i |
3 | -0.4537 + 0.5765 i | 7.09e-9 + 1.93e-9 i |
4 | -0.4537 - 0.5765 i | 7.09e-9 - 1.93e-9 i |
5 | -0.7491 + 0.1846 i | 2.6e-9 + 5.15e-9 i |
6 | -0.7491 - 0.1846 i | 2.6e-9 - 5.15e-9 i |
7 | -0.7501 + 0.1845 i | 5.17e-9 - 2.59e-9 i |
8 | -0.7501 - 0.1845 i | 5.17e-9 + 2.59e-9 i |
9 | -0.0174 + 0.831 i | 4.5e-8 + 4.91e-8 i |
10 | -0.0174 - 0.831 i | 4.5e-8 - 4.91e-8 i |
11 | -0.0172 + 0.8317 i | 4.94e-8 - 4.51e-8 i |
12 | -0.0172 - 0.8317 i | 4.94e-8 + 4.51e-8 i |
13 | 1.254 + 0.2117 i | 0.000287 + 0.000383 i |
14 | 1.254 - 0.2117 i | 0.000287 - 0.000383 i |
15 | 1.3451 + 0.3132 i | 0.00108 + 0.000305 i |
16 | 1.3451 - 0.3132 i | 0.00108 - 0.000305 i |
17 | -2.3386 + 1.6715 i | 0.000163 - 0.000156 i |
18 | -2.3386 - 1.6715 i | 0.000163 + 0.000156 i |
19 | 3.382 | 0.565 |
20 | 19.704 | 2.22 i |
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Table 14. Singularities with their weights for the quadratic approximant [10, 10, 10] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.3427 + 0.7428 i | 4.7e-9 + 3.51e-8 i |
![Singularities of quadratic [10, 10, 10] approximant](singsq14.gif?312172) |
2 | -0.3427 - 0.7428 i | 4.7e-9 - 3.51e-8 i |
3 | -0.3426 + 0.7436 i | 3.51e-8 - 4.71e-9 i |
4 | -0.3426 - 0.7436 i | 3.51e-8 + 4.71e-9 i |
5 | 0.0856 + 0.9185 i | 1.95e-7 - 9.24e-9 i |
6 | 0.0856 - 0.9185 i | 1.95e-7 + 9.24e-9 i |
7 | 0.087 + 0.9184 i | 1.03e-8 + 1.96e-7 i |
8 | 0.087 - 0.9184 i | 1.03e-8 - 1.96e-7 i |
9 | -1.0382 + 0.3901 i | 1.84e-7 + 7.37e-9 i |
10 | -1.0382 - 0.3901 i | 1.84e-7 - 7.37e-9 i |
11 | -1.0444 + 0.4135 i | 1.07e-9 + 1.93e-7 i |
12 | -1.0444 - 0.4135 i | 1.07e-9 - 1.93e-7 i |
13 | 1.3528 + 0.2502 i | 0.00043 - 0.00195 i |
14 | 1.3528 - 0.2502 i | 0.00043 + 0.00195 i |
15 | 1.3651 + 0.435 i | 0.000519 - 0.0016 i |
16 | 1.3651 - 0.435 i | 0.000519 + 0.0016 i |
17 | -1.6308 + 1.3208 i | 0.0000166 - 0.0000187 i |
18 | -1.6308 - 1.3208 i | 0.0000166 + 0.0000187 i |
19 | 4.7439 | 1.59 |
20 | -5.2008 | 0.106 |
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Table 15. Singularities with their weights for the quadratic approximant [11, 10, 10] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.0283 | 0 |
![Singularities of quadratic [11, 10, 10] approximant](singsq15.gif?235603) |
2 | 0.0283 | 0 |
3 | -0.3549 + 0.7407 i | 7.32e-9 + 2.02e-8 i |
4 | -0.3549 - 0.7407 i | 7.32e-9 - 2.02e-8 i |
5 | -0.3552 + 0.7418 i | 2.02e-8 - 7.31e-9 i |
6 | -0.3552 - 0.7418 i | 2.02e-8 + 7.31e-9 i |
7 | 0.0731 + 0.9268 i | 1.32e-7 - 4.84e-8 i |
8 | 0.0731 - 0.9268 i | 1.32e-7 + 4.84e-8 i |
9 | 0.0751 + 0.9275 i | 4.94e-8 + 1.33e-7 i |
10 | 0.0751 - 0.9275 i | 4.94e-8 - 1.33e-7 i |
11 | -1.0067 + 0.3618 i | 8.98e-8 + 3.4e-9 i |
12 | -1.0067 - 0.3618 i | 8.98e-8 - 3.4e-9 i |
13 | -1.0182 + 0.3815 i | 9.57e-8 i |
14 | -1.0182 - 0.3815 i | -9.57e-8 i |
15 | 1.3693 + 0.2535 i | 0.000657 - 0.00253 i |
16 | 1.3693 - 0.2535 i | 0.000657 + 0.00253 i |
17 | 1.3607 + 0.4401 i | 0.000453 - 0.00161 i |
18 | 1.3607 - 0.4401 i | 0.000453 + 0.00161 i |
19 | -1.5381 + 1.5761 i | 9.44e-6 + 0.0000237 i |
20 | -1.5381 - 1.5761 i | 9.44e-6 - 0.0000237 i |
21 | 14.2055 | 0.155 |
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Table 16. Singularities with their weights for the quadratic approximant [11, 11, 10] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.8654 + 0.1615 i | 3.92e-8 + 5.06e-8 i |
![Singularities of quadratic [11, 11, 10] approximant](singsq16.gif?130352) |
2 | -0.8654 - 0.1615 i | 3.92e-8 - 5.06e-8 i |
3 | -0.8663 + 0.1625 i | 5.09e-8 - 3.93e-8 i |
4 | -0.8663 - 0.1625 i | 5.09e-8 + 3.93e-8 i |
5 | -0.4731 + 0.7593 i | 9.99e-8 - 7.08e-8 i |
6 | -0.4731 - 0.7593 i | 9.99e-8 + 7.08e-8 i |
7 | -0.4722 + 0.7605 i | 7.1e-8 + 9.98e-8 i |
8 | -0.4722 - 0.7605 i | 7.1e-8 - 9.98e-8 i |
9 | -0.063 + 0.993 i | 4.13e-7 - 8.85e-8 i |
10 | -0.063 - 0.993 i | 4.13e-7 + 8.85e-8 i |
11 | -0.066 + 0.9953 i | 8.3e-8 + 4.13e-7 i |
12 | -0.066 - 0.9953 i | 8.3e-8 - 4.13e-7 i |
13 | 1.3754 + 0.5234 i | 0.00198 - 0.000523 i |
14 | 1.3754 - 0.5234 i | 0.00198 + 0.000523 i |
15 | 1.5194 + 0.2476 i | 0.0201 - 0.00555 i |
16 | 1.5194 - 0.2476 i | 0.0201 + 0.00555 i |
17 | 0.618 + 1.8043 i | 0.0000661 + 0.0000298 i |
18 | 0.618 - 1.8043 i | 0.0000661 - 0.0000298 i |
19 | 0.368 + 1.8769 i | 0.0000438 - 0.0000228 i |
20 | 0.368 - 1.8769 i | 0.0000438 + 0.0000228 i |
21 | -2.4905 | 0.000475 |
22 | -6.4821 | 0.0147 i |
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Table 17. Singularities with their weights for the quadratic approximant [11, 11, 11] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.7353 | 6.02e-10 |
![Singularities of quadratic [11, 11, 11] approximant](singsq17.gif?715267) |
2 | -0.7361 | 6.01e-10 i |
3 | -0.46 + 0.757 i | 1.01e-8 - 7.35e-9 i |
4 | -0.46 - 0.757 i | 1.01e-8 + 7.35e-9 i |
5 | -0.4556 + 0.7608 i | 7.56e-9 + 1.01e-8 i |
6 | -0.4556 - 0.7608 i | 7.56e-9 - 1.01e-8 i |
7 | -0.0479 + 0.9828 i | 6.35e-8 - 3.57e-8 i |
8 | -0.0479 - 0.9828 i | 6.35e-8 + 3.57e-8 i |
9 | -0.0562 + 0.9891 i | 3.2e-8 + 6.44e-8 i |
10 | -0.0562 - 0.9891 i | 3.2e-8 - 6.44e-8 i |
11 | -0.9872 + 0.2184 i | 8.52e-9 + 4.97e-9 i |
12 | -0.9872 - 0.2184 i | 8.52e-9 - 4.97e-9 i |
13 | -1.0264 + 0.1929 i | 7.89e-9 - 7.44e-9 i |
14 | -1.0264 - 0.1929 i | 7.89e-9 + 7.44e-9 i |
15 | 1.3634 + 0.5083 i | 0.00104 - 0.00137 i |
16 | 1.3634 - 0.5083 i | 0.00104 + 0.00137 i |
17 | 1.6286 + 0.1791 i | 0.208 + 0.1 i |
18 | 1.6286 - 0.1791 i | 0.208 - 0.1 i |
19 | 0.8234 + 1.6764 i | 0.0000393 - 0.0000161 i |
20 | 0.8234 - 1.6764 i | 0.0000393 + 0.0000161 i |
21 | 0.3448 + 2.1328 i | 0.0000295 + 5.93e-6 i |
22 | 0.3448 - 2.1328 i | 0.0000295 - 5.93e-6 i |
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Table 18. Singularities with their weights for the quadratic approximant [12, 11, 11] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.3742 + 0.e-5 i | 0 |
![Singularities of quadratic [12, 11, 11] approximant](singsq18.gif?538296) |
2 | 0.3742 - 0.e-5 i | 0 |
3 | -0.8607 + 0.163 i | 5.e-10 + 5.e-9 i |
4 | -0.8607 - 0.163 i | 5.e-10 - 5.e-9 i |
5 | -0.8655 + 0.1575 i | 5.01e-9 - 3.42e-10 i |
6 | -0.8655 - 0.1575 i | 5.01e-9 + 3.42e-10 i |
7 | -0.4728 + 0.7646 i | 6.21e-9 - 8.98e-9 i |
8 | -0.4728 - 0.7646 i | 6.21e-9 + 8.98e-9 i |
9 | -0.4812 + 0.7631 i | 8.91e-9 + 6.46e-9 i |
10 | -0.4812 - 0.7631 i | 8.91e-9 - 6.46e-9 i |
11 | -0.0648 + 0.9977 i | 3.96e-8 + 2.41e-8 i |
12 | -0.0648 - 0.9977 i | 3.96e-8 - 2.41e-8 i |
13 | -0.0739 + 1.0164 i | 2.82e-8 - 3.85e-8 i |
14 | -0.0739 - 1.0164 i | 2.82e-8 + 3.85e-8 i |
15 | 1.3362 + 0.4845 i | 0.000113 - 0.000837 i |
16 | 1.3362 - 0.4845 i | 0.000113 + 0.000837 i |
17 | 1.5977 + 0.127 i | 0.731 + 0.425 i |
18 | 1.5977 - 0.127 i | 0.731 - 0.425 i |
19 | 0.0672 + 1.6865 i | 1.53e-6 + 2.32e-6 i |
20 | 0.0672 - 1.6865 i | 1.53e-6 - 2.32e-6 i |
21 | 0.4688 + 1.6413 i | 3.7e-6 + 6.05e-6 i |
22 | 0.4688 - 1.6413 i | 3.7e-6 - 6.05e-6 i |
23 | -5.4507 | 0.00931 |
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Table 19. Singularities with their weights for the quadratic approximant [12, 12, 11] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.0273 + 0.6909 i | 1.62e-10 - 1.1e-10 i |
![Singularities of quadratic [12, 12, 11] approximant](singsq19.gif?755715) |
2 | -0.0273 - 0.6909 i | 1.62e-10 + 1.1e-10 i |
3 | -0.0271 + 0.691 i | 1.1e-10 + 1.62e-10 i |
4 | -0.0271 - 0.691 i | 1.1e-10 - 1.62e-10 i |
5 | -0.4891 + 0.7283 i | 1.96e-9 + 1.9e-10 i |
6 | -0.4891 - 0.7283 i | 1.96e-9 - 1.9e-10 i |
7 | -0.8582 + 0.196 i | 9.65e-10 + 1.5e-9 i |
8 | -0.8582 - 0.196 i | 9.65e-10 - 1.5e-9 i |
9 | -0.4791 + 0.7417 i | -1.99e-9 i |
10 | -0.4791 - 0.7417 i | 1.99e-9 i |
11 | -0.8718 + 0.1882 i | 1.62e-9 - 9.01e-10 i |
12 | -0.8718 - 0.1882 i | 1.62e-9 + 9.01e-10 i |
13 | -0.0799 + 0.9297 i | 3.79e-9 + 1.73e-9 i |
14 | -0.0799 - 0.9297 i | 3.79e-9 - 1.73e-9 i |
15 | -0.0504 + 0.9351 i | 1.01e-9 - 4.6e-9 i |
16 | -0.0504 - 0.9351 i | 1.01e-9 + 4.6e-9 i |
17 | 1.3175 + 0.2559 i | 0.000351 - 0.000169 i |
18 | 1.3175 - 0.2559 i | 0.000351 + 0.000169 i |
19 | 1.2223 + 0.7092 i | 0.000032 - 0.0000304 i |
20 | 1.2223 - 0.7092 i | 0.000032 + 0.0000304 i |
21 | 1.2252 + 1.122 i | 0.0000291 + 0.0000194 i |
22 | 1.2252 - 1.122 i | 0.0000291 - 0.0000194 i |
23 | 0.5437 + 1.8375 i | 6.59e-6 + 4.29e-6 i |
24 | 0.5437 - 1.8375 i | 6.59e-6 - 4.29e-6 i |
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Table 20. Singularities with their weights for the quadratic approximant [12, 12, 12] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.2501 + 0.703 i | 0 |
![Singularities of quadratic [12, 12, 12] approximant](singsq20.gif?403624) |
2 | -0.2501 - 0.703 i | 0 |
3 | -0.2485 + 0.7052 i | -1.e-10 i |
4 | -0.2485 - 0.7052 i | 1.e-10 i |
5 | -0.477 + 0.6942 i | 3.07e-10 i |
6 | -0.477 - 0.6942 i | -3.07e-10 i |
7 | -0.501 + 0.7178 i | 3.68e-10 |
8 | -0.501 - 0.7178 i | 3.68e-10 |
9 | -0.8571 + 0.1924 i | 3.15e-10 |
10 | -0.8571 - 0.1924 i | 3.15e-10 |
11 | -0.8606 + 0.2389 i | -3.62e-10 i |
12 | -0.8606 - 0.2389 i | 3.62e-10 i |
13 | -0.0337 + 0.9375 i | 2.39e-9 + 1.83e-9 i |
14 | -0.0337 - 0.9375 i | 2.39e-9 - 1.83e-9 i |
15 | -0.0256 + 0.9846 i | 3.2e-9 - 2.65e-9 i |
16 | -0.0256 - 0.9846 i | 3.2e-9 + 2.65e-9 i |
17 | 1.3404 | 0.00083 |
18 | 1.2544 + 0.474 i | 0.0000498 + 0.00011 i |
19 | 1.2544 - 0.474 i | 0.0000498 - 0.00011 i |
20 | -1.4907 + 0.3006 i | 9.11e-9 - 3.21e-8 i |
21 | -1.4907 - 0.3006 i | 9.11e-9 + 3.21e-8 i |
22 | 0.603 + 1.4707 i | 1.66e-6 + 8.68e-7 i |
23 | 0.603 - 1.4707 i | 1.66e-6 - 8.68e-7 i |
24 | 3.9474 | 0.102 i |
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Examples | a1 | a2 | a8 | a16 | a22 | a30 | a38 | a44 | a45 | a51 | a62 | a69 | a75 | a83 | a84 | a85 | a86 | a87 | a88 | a90 | a91 |
Molecule | Ar | BH | BH | BH | BH | BH | BH | BO+ | C2 | CN+ | N2 | HF | HF | HCl | HCl | F- | Cl- | Cl- | Ne | OH- | SH- |
Basis | aug-cc-pVDZ | cc-pVDZ | cc-pVTZ | cc-pVQZ | aug-cc-pVDZ | aug-cc-pVTZ | aug-cc-pVQZ | cc-pVDZ | cc-pVDZ | cc-pVDZ | cc-pVDZ | cc-pVDZ | aug-cc-pVDZ | cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ | cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ | aug-cc-pVDZ |
Designed by A. Sergeev.