Singularities of Møller-Plesset series: example "o1"
Molecule O2-. Basis aug-cc-pVDZ. Structure ""
Content
- Definition of quadratic approximants
- Approximant [2, 2, 2]
- Approximant [2, 2, 3]
- Approximant [2, 3, 3]
- Approximant [3, 3, 3]
- Approximant [3, 3, 4]
- Approximant [3, 4, 4]
- Approximant [4, 4, 4]
- Approximant [4, 4, 5]
- Approximant [4, 5, 5]
- Approximant [5, 5, 5]
- Approximant [5, 5, 6]
- Approximant [5, 6, 6]
- Approximant [6, 6, 6]
- Approximant [6, 6, 7]
- Approximant [6, 7, 7]
Examples | o1 |
Molecule | O2- |
Basis | 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 [2, 2, 2] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5863 + 0.0429 i | 0.605 + 1.34 i |
![Singularities of quadratic [2, 2, 2] approximant](singsq1.gif?25008) |
2 | -0.5863 - 0.0429 i | 0.605 - 1.34 i |
3 | -1.2334 | 0.77 |
4 | 2.4084 | 2.34 |
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Table 2. Singularities with their weights for the quadratic approximant [2, 2, 3] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5652 + 0.0714 i | 0.207 + 0.552 i |
![Singularities of quadratic [2, 2, 3] approximant](singsq2.gif?928633) |
2 | -0.5652 - 0.0714 i | 0.207 - 0.552 i |
3 | -1.0352 | 0.396 |
4 | 2.9253 | 30.8 |
5 | 7.1608 | 1.32 i |
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Table 3. Singularities with their weights for the quadratic approximant [2, 3, 3] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5867 | 17.3 |
![Singularities of quadratic [2, 3, 3] approximant](singsq3.gif?996106) |
2 | -0.6228 | 3.16 i |
3 | 1.9657 | 0.691 |
4 | -2.0289 | 326. |
5 | -3.017 | 1.68 i |
6 | 403.786 | 11.2 i |
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Table 4. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5917 + 0.0384 i | 1.16 + 1.44 i |
![Singularities of quadratic [3, 3, 3] approximant](singsq4.gif?905674) |
2 | -0.5917 - 0.0384 i | 1.16 - 1.44 i |
3 | 0.8645 | 0.108 |
4 | 0.8811 | 0.109 i |
5 | -1.3193 | 1.1 |
6 | 2.7423 | 8.57 |
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Table 5. Singularities with their weights for the quadratic approximant [3, 3, 4] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5808 + 0.0633 i | 0.649 + 0.434 i |
![Singularities of quadratic [3, 3, 4] approximant](singsq5.gif?766308) |
2 | -0.5808 - 0.0633 i | 0.649 - 0.434 i |
3 | -0.9319 | 0.624 |
4 | -1.6087 | 0.354 i |
5 | 1.7353 | 0.348 |
6 | -0.9951 + 1.6883 i | 0.367 - 0.182 i |
7 | -0.9951 - 1.6883 i | 0.367 + 0.182 i |
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Table 6. Singularities with their weights for the quadratic approximant [3, 4, 4] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.362 | 0.000328 |
![Singularities of quadratic [3, 4, 4] approximant](singsq6.gif?61912) |
2 | 0.3623 | 0.000328 i |
3 | -0.4648 + 0.0134 i | 0.0166 + 0.013 i |
4 | -0.4648 - 0.0134 i | 0.0166 - 0.013 i |
5 | -0.8439 | 0.0915 |
6 | 1.3718 | 0.0536 |
7 | -2.2046 | 0.799 i |
8 | 5.8892 | 3.94 i |
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Table 7. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5504 + 0.1338 i | 0.0687 - 0.0038 i |
![Singularities of quadratic [4, 4, 4] approximant](singsq7.gif?485789) |
2 | -0.5504 - 0.1338 i | 0.0687 + 0.0038 i |
3 | -0.629 + 0.1654 i | 0.0253 + 0.0703 i |
4 | -0.629 - 0.1654 i | 0.0253 - 0.0703 i |
5 | -1.4604 | 0.433 |
6 | 1.5615 | 0.183 |
7 | 3.7696 | 0.661 i |
8 | -4.9544 | 1.59 i |
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Table 8. Singularities with their weights for the quadratic approximant [4, 4, 5] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5715 + 0.0931 i | 0.239 + 0.0152 i |
![Singularities of quadratic [4, 4, 5] approximant](singsq8.gif?531928) |
2 | -0.5715 - 0.0931 i | 0.239 - 0.0152 i |
3 | -0.7444 + 0.1046 i | 0.184 + 0.232 i |
4 | -0.7444 - 0.1046 i | 0.184 - 0.232 i |
5 | 1.5379 | 0.201 |
6 | -2.1686 + 0.9353 i | 1.66 + 0.224 i |
7 | -2.1686 - 0.9353 i | 1.66 - 0.224 i |
8 | 2.3906 | 0.355 i |
9 | 4.3953 | 4.93 |
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Table 9. Singularities with their weights for the quadratic approximant [4, 5, 5] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5942 | 0.375 |
![Singularities of quadratic [4, 5, 5] approximant](singsq9.gif?416598) |
2 | -0.592 + 0.0723 i | 0.0451 + 0.327 i |
3 | -0.592 - 0.0723 i | 0.0451 - 0.327 i |
4 | -0.4983 + 0.5501 i | 0.00797 - 0.00501 i |
5 | -0.4983 - 0.5501 i | 0.00797 + 0.00501 i |
6 | -0.4912 + 0.5781 i | 0.0053 + 0.00794 i |
7 | -0.4912 - 0.5781 i | 0.0053 - 0.00794 i |
8 | -1.4804 | 0.149 i |
9 | 1.4925 | 0.106 |
10 | 7.7509 | 12.3 i |
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Table 10. Singularities with their weights for the quadratic approximant [5, 5, 5] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.3452 | 0.0000641 |
![Singularities of quadratic [5, 5, 5] approximant](singsq10.gif?63067) |
2 | 0.3453 | 0.0000641 i |
3 | -0.4841 + 0.0164 i | 0.0093 + 0.00639 i |
4 | -0.4841 - 0.0164 i | 0.0093 - 0.00639 i |
5 | -0.5782 + 0.1969 i | 0.0225 + 0.0126 i |
6 | -0.5782 - 0.1969 i | 0.0225 - 0.0126 i |
7 | -0.6505 | 0.668 |
8 | -0.8653 | 0.0528 i |
9 | 1.414 | 0.0546 |
10 | -25.9719 | 0.457 |
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Table 11. Singularities with their weights for the quadratic approximant [5, 5, 6] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5687 + 0.0692 i | 0.377 + 0.417 i |
![Singularities of quadratic [5, 5, 6] approximant](singsq11.gif?693549) |
2 | -0.5687 - 0.0692 i | 0.377 - 0.417 i |
3 | -0.8341 | 2.45 |
4 | -0.8763 | 19.5 i |
5 | -1.4684 + 1.0037 i | 0.307 + 0.668 i |
6 | -1.4684 - 1.0037 i | 0.307 - 0.668 i |
7 | 1.9305 + 0.5516 i | 0.277 + 0.0516 i |
8 | 1.9305 - 0.5516 i | 0.277 - 0.0516 i |
9 | 1.5205 + 1.3753 i | 0.0955 - 0.205 i |
10 | 1.5205 - 1.3753 i | 0.0955 + 0.205 i |
11 | 3.5009 | 0.393 |
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Table 12. Singularities with their weights for the quadratic approximant [5, 6, 6] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5696 + 0.0698 i | 0.435 + 0.386 i |
![Singularities of quadratic [5, 6, 6] approximant](singsq12.gif?347520) |
2 | -0.5696 - 0.0698 i | 0.435 - 0.386 i |
3 | -0.7769 | 0.652 |
4 | -0.9631 | 0.56 i |
5 | 1.5441 + 0.4115 i | 0.0904 - 0.105 i |
6 | 1.5441 - 0.4115 i | 0.0904 + 0.105 i |
7 | -1.3204 + 1.2207 i | 0.445 + 0.107 i |
8 | -1.3204 - 1.2207 i | 0.445 - 0.107 i |
9 | 2.0508 + 1.018 i | 0.171 + 0.144 i |
10 | 2.0508 - 1.018 i | 0.171 - 0.144 i |
11 | -1.3832 + 6.1748 i | 0.638 + 0.473 i |
12 | -1.3832 - 6.1748 i | 0.638 - 0.473 i |
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Table 13. 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 | -0.5734 + 0.0706 i | 0.671 + 0.209 i |
![Singularities of quadratic [6, 6, 6] approximant](singsq13.gif?165714) |
2 | -0.5734 - 0.0706 i | 0.671 - 0.209 i |
3 | -0.7177 | 0.335 |
4 | 1.2516 | 0.0116 |
5 | -1.2964 | 0.116 i |
6 | 1.2896 + 0.6064 i | 0.0106 + 0.0255 i |
7 | 1.2896 - 0.6064 i | 0.0106 - 0.0255 i |
8 | -1.6548 | 0.176 |
9 | 1.6722 | 0.0089 i |
10 | 1.767 | 0.0116 |
11 | -1.2203 + 1.3043 i | 0.156 - 0.223 i |
12 | -1.2203 - 1.3043 i | 0.156 + 0.223 i |
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Table 14. Singularities with their weights for the quadratic approximant [6, 6, 7] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5665 + 0.0695 i | 0.25 + 0.407 i |
![Singularities of quadratic [6, 6, 7] approximant](singsq14.gif?99248) |
2 | -0.5665 - 0.0695 i | 0.25 - 0.407 i |
3 | -0.7731 + 0.1038 i | 0.027 - 0.25 i |
4 | -0.7731 - 0.1038 i | 0.027 + 0.25 i |
5 | -0.9073 | 0.188 |
6 | -1.2048 | 0.597 i |
7 | -1.3706 + 0.8677 i | 0.253 - 0.577 i |
8 | -1.3706 - 0.8677 i | 0.253 + 0.577 i |
9 | 1.5913 + 0.4808 i | 0.116 - 0.0438 i |
10 | 1.5913 - 0.4808 i | 0.116 + 0.0438 i |
11 | 1.7511 + 1.3545 i | 0.0429 - 0.185 i |
12 | 1.7511 - 1.3545 i | 0.0429 + 0.185 i |
13 | 52.4936 | 6.22 |
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Table 15. Singularities with their weights for the quadratic approximant [6, 7, 7] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5716 + 0.0739 i | 0.457 + 0.138 i |
![Singularities of quadratic [6, 7, 7] approximant](singsq15.gif?813416) |
2 | -0.5716 - 0.0739 i | 0.457 - 0.138 i |
3 | -0.6343 | 0.624 |
4 | -0.6832 | 0.514 i |
5 | -0.8671 + 0.1302 i | 0.288 + 0.167 i |
6 | -0.8671 - 0.1302 i | 0.288 - 0.167 i |
7 | 1.6233 + 0.4488 i | 0.139 - 0.122 i |
8 | 1.6233 - 0.4488 i | 0.139 + 0.122 i |
9 | -1.3749 + 1.0848 i | 0.345 + 0.32 i |
10 | -1.3749 - 1.0848 i | 0.345 - 0.32 i |
11 | 1.8232 + 1.1543 i | 0.115 + 0.207 i |
12 | 1.8232 - 1.1543 i | 0.115 - 0.207 i |
13 | 4.4191 + 10.3127 i | 0.969 + 0.115 i |
14 | 4.4191 - 10.3127 i | 0.969 - 0.115 i |
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Examples | o1 |
Molecule | O2- |
Basis | aug-cc-pVDZ |
Designed by A. Sergeev.