Singularities of Møller-Plesset series: example "a44"
Molecule BO+. Basis cc-pVDZ. Structure "mpn_Rfci"
Content
- Definition of quadratic approximants
- Approximant [1, 0, 0]
- Approximant [1, 1, 0]
- Approximant [1, 1, 1]
- Approximant [2, 1, 1]
- Approximant [2, 2, 1]
- Approximant [2, 2, 2]
- Approximant [3, 2, 2]
- Approximant [3, 3, 2]
- Approximant [3, 3, 3]
- Approximant [4, 3, 3]
- Approximant [4, 4, 3]
- Approximant [4, 4, 4]
- Approximant [5, 4, 4]
- Approximant [5, 5, 4]
- Approximant [5, 5, 5]
- Approximant [6, 5, 5]
- Approximant [6, 6, 5]
- Approximant [6, 6, 6]
- Approximant [7, 6, 6]
- Approximant [7, 7, 6]
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 [1, 0, 0] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -2.8519 | 1.55 |
![Singularities of quadratic [1, 0, 0] approximant](singsq1.gif?685976) |
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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.3684 | 0.0571 |
![Singularities of quadratic [1, 1, 0] approximant](singsq2.gif?674370) |
2 | -0.8976 | 0.0892 i |
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Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.6746 | 0.212 |
![Singularities of quadratic [1, 1, 1] approximant](singsq3.gif?830746) |
2 | 8.4475 | 1.57 |
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Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.4915 | 0.0542 |
![Singularities of quadratic [2, 1, 1] approximant](singsq4.gif?935237) |
2 | 0.7452 + 1.0275 i | 0.103 + 0.00756 i |
3 | 0.7452 - 1.0275 i | 0.103 - 0.00756 i |
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Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5089 | 0.0646 |
![Singularities of quadratic [2, 2, 1] approximant](singsq5.gif?499730) |
2 | 0.9609 + 1.0451 i | 0.135 + 0.0076 i |
3 | 0.9609 - 1.0451 i | 0.135 - 0.0076 i |
4 | -64.8418 | 3.84 i |
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Table 6. 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 | 0.0719 | 0.0000567 |
![Singularities of quadratic [2, 2, 2] approximant](singsq6.gif?566436) |
2 | 0.0719 | 0.0000567 i |
3 | -0.4707 | 0.0329 |
4 | 1.2306 | 0.121 |
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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | 0.2387 + 0.0002 i | 0.000526 - 0.000525 i |
![Singularities of quadratic [3, 2, 2] approximant](singsq7.gif?569123) |
2 | 0.2387 - 0.0002 i | 0.000526 + 0.000525 i |
3 | -0.4833 | 0.0419 |
4 | 1.5107 | 0.236 |
5 | 4.3914 | 4.36 i |
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Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.4859 | 0.0448 |
![Singularities of quadratic [3, 3, 2] approximant](singsq8.gif?706981) |
2 | 0.5252 + 0.0064 i | 0.00309 - 0.00305 i |
3 | 0.5252 - 0.0064 i | 0.00309 + 0.00305 i |
4 | 1.3395 | 0.0918 |
5 | 3.9881 | 90.3 i |
6 | -140.185 | 13.2 i |
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Table 9. 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 | -0.4867 | 0.0479 |
![Singularities of quadratic [3, 3, 3] approximant](singsq9.gif?839803) |
2 | -0.1053 + 0.6566 i | 0.00849 - 0.0148 i |
3 | -0.1053 - 0.6566 i | 0.00849 + 0.0148 i |
4 | -0.128 + 0.6598 i | 0.0144 + 0.00884 i |
5 | -0.128 - 0.6598 i | 0.0144 - 0.00884 i |
6 | 1.388 | 0.273 |
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Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.3605 + 0.0006 i | 0.00188 + 0.00189 i |
![Singularities of quadratic [4, 3, 3] approximant](singsq10.gif?721144) |
2 | -0.3605 - 0.0006 i | 0.00188 - 0.00189 i |
3 | -0.4733 | 0.0252 |
4 | 1.3289 | 0.286 |
5 | 0.0714 + 2.1233 i | 0.0207 + 0.27 i |
6 | 0.0714 - 2.1233 i | 0.0207 - 0.27 i |
7 | -14.3235 | 1.2 i |
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Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.0406 | 2.41e-7 |
![Singularities of quadratic [4, 4, 3] approximant](singsq11.gif?669306) |
2 | -0.0406 | 2.41e-7 i |
3 | -0.4755 | 0.0317 |
4 | 1.2025 | 0.0986 |
5 | -1.5307 | 0.507 i |
6 | -1.9408 + 1.7912 i | 0.409 + 0.863 i |
7 | -1.9408 - 1.7912 i | 0.409 - 0.863 i |
8 | 11.0478 | 499. i |
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Table 12. 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 | -0.502 | 0.131 |
![Singularities of quadratic [4, 4, 4] approximant](singsq12.gif?958322) |
2 | -0.5608 | 0.331 i |
3 | -0.6432 | 0.263 |
4 | -1.1901 | 0.264 i |
5 | 1.2658 + 0.1781 i | 0.109 - 0.102 i |
6 | 1.2658 - 0.1781 i | 0.109 + 0.102 i |
7 | 1.7864 | 0.285 |
8 | -2.2802 | 832. |
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Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5093 | 0.219 |
![Singularities of quadratic [5, 4, 4] approximant](singsq13.gif?769771) |
2 | -0.5399 | 1.58 i |
3 | -0.6224 | 0.171 |
4 | -1.2326 | 0.261 i |
5 | 1.2629 + 0.1703 i | 0.108 - 0.106 i |
6 | 1.2629 - 0.1703 i | 0.108 + 0.106 i |
7 | 1.752 | 0.27 |
8 | -2.488 | 97.7 |
9 | -34.9609 | 0.883 i |
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Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5158 + 0.0348 i | 0.089 + 0.0382 i |
![Singularities of quadratic [5, 5, 4] approximant](singsq14.gif?355955) |
2 | -0.5158 - 0.0348 i | 0.089 - 0.0382 i |
3 | -0.6168 | 0.238 |
4 | -0.4255 + 0.6014 i | 0.00521 - 0.00544 i |
5 | -0.4255 - 0.6014 i | 0.00521 + 0.00544 i |
6 | -0.4137 + 0.614 i | 0.00554 + 0.00507 i |
7 | -0.4137 - 0.614 i | 0.00554 - 0.00507 i |
8 | 1.1637 | 0.0781 |
9 | 4.9348 | 7.03 i |
10 | -5.2832 | 9.18 i |
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Table 15. 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 | 0.1005 | 3.23e-7 |
![Singularities of quadratic [5, 5, 5] approximant](singsq15.gif?364414) |
2 | 0.1005 | 3.23e-7 i |
3 | -0.5172 | 1.08 |
4 | 0.5312 | 0.00045 |
5 | 0.5327 | 0.000451 i |
6 | -0.5384 | 0.643 i |
7 | -0.6385 | 0.336 |
8 | 1.0829 | 0.0279 |
9 | -1.5288 | 0.196 i |
10 | -4.6261 | 0.898 |
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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5] The most stable singularity is highlighted. |
No. | zc | c | Location in the complex plane |
1 | -0.5163 + 0.0213 i | 0.0427 + 0.212 i |
![Singularities of quadratic [6, 5, 5] approximant](singsq16.gif?832522) |
2 | -0.5163 - 0.0213 i | 0.0427 - 0.212 i |
3 | -0.5987 | 0.114 |
4 | 0.9051 + 0.7127 i | 0.0136 - 0.0259 i |
5 | 0.9051 - 0.7127 i | 0.0136 + 0.0259 i |
6 | 1.1611 | 0.0685 |
7 | 0.8977 + 0.7623 i | 0.0273 + 0.0151 i |
8 | 0.8977 - 0.7623 i | 0.0273 - 0.0151 i |
9 | -1.4093 | 0.223 i |
10 | -4.2368 + 2.6667 i | 0.288 + 0.732 i |
11 | -4.2368 - 2.6667 i | 0.288 - 0.732 i |
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Table 17. 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 | 0.4349 + 0.e-4 i | 0.000137 - 0.000137 i |
![Singularities of quadratic [6, 6, 5] approximant](singsq17.gif?708533) |
2 | 0.4349 - 0.e-4 i | 0.000137 + 0.000137 i |
3 | -0.5206 + 0.0187 i | 0.165 + 0.25 i |
4 | -0.5206 - 0.0187 i | 0.165 - 0.25 i |
5 | -0.6158 | 0.176 |
6 | 1.4456 + 0.1362 i | 0.161 + 0.238 i |
7 | 1.4456 - 0.1362 i | 0.161 - 0.238 i |
8 | -1.8967 | 0.196 i |
9 | -0.3142 + 2.2012 i | 0.0564 + 0.0849 i |
10 | -0.3142 - 2.2012 i | 0.0564 - 0.0849 i |
11 | 1.1813 + 2.0801 i | 0.0122 - 0.113 i |
12 | 1.1813 - 2.0801 i | 0.0122 + 0.113 i |
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Table 18. 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 | 0.1106 | 1.07e-7 |
![Singularities of quadratic [6, 6, 6] approximant](singsq18.gif?374830) |
2 | 0.1106 | 1.07e-7 i |
3 | -0.5234 + 0.0173 i | 0.293 + 0.195 i |
4 | -0.5234 - 0.0173 i | 0.293 - 0.195 i |
5 | -0.6275 | 0.266 |
6 | 1.1932 + 0.2526 i | 0.0546 - 0.023 i |
7 | 1.1932 - 0.2526 i | 0.0546 + 0.023 i |
8 | -1.3844 + 1.09 i | 0.0607 + 0.105 i |
9 | -1.3844 - 1.09 i | 0.0607 - 0.105 i |
10 | -0.9386 + 1.5147 i | 0.0386 - 0.108 i |
11 | -0.9386 - 1.5147 i | 0.0386 + 0.108 i |
12 | 1.7889 | 0.22 |
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Table 19. 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 | -0.523 + 0.0132 i | 0.192 + 0.471 i |
![Singularities of quadratic [7, 6, 6] approximant](singsq19.gif?539723) |
2 | -0.523 - 0.0132 i | 0.192 - 0.471 i |
3 | -0.6467 | 0.3 |
4 | -0.7566 | 1.72 i |
5 | -0.8581 | 0.49 |
6 | 1.2494 + 0.2224 i | 0.0866 - 0.089 i |
7 | 1.2494 - 0.2224 i | 0.0866 + 0.089 i |
8 | -1.2847 | 0.368 i |
9 | 1.9754 + 0.5013 i | 0.299 + 0.148 i |
10 | 1.9754 - 0.5013 i | 0.299 - 0.148 i |
11 | -1.8867 + 1.5953 i | 0.881 + 0.0895 i |
12 | -1.8867 - 1.5953 i | 0.881 - 0.0895 i |
13 | 6.9332 | 0.957 |
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Table 20. 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 | -0.5227 + 0.0131 i | 0.142 + 0.486 i |
![Singularities of quadratic [7, 7, 6] approximant](singsq20.gif?163897) |
2 | -0.5227 - 0.0131 i | 0.142 - 0.486 i |
3 | -0.6463 | 0.274 |
4 | -0.7442 | 4.4 i |
5 | -0.8385 | 0.373 |
6 | 1.2482 + 0.2268 i | 0.087 - 0.0774 i |
7 | 1.2482 - 0.2268 i | 0.087 + 0.0774 i |
8 | -1.301 | 0.381 i |
9 | 2.0444 + 0.554 i | 0.303 + 0.254 i |
10 | 2.0444 - 0.554 i | 0.303 - 0.254 i |
11 | -1.9223 + 1.4827 i | 1.02 + 0.481 i |
12 | -1.9223 - 1.4827 i | 1.02 - 0.481 i |
13 | 11.4624 | 2.14 |
14 | 86.4314 | 73.7 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.