Singularities of Møller-Plesset series: example "H--cc-pVQZ"
Molecule H- ion. Basis AUG-CC-PVQZ. 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]
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 | 1.3885 | 0.0483 |
![Singularities of quadratic [2, 2, 2] approximant](singsq1.gif?130712) |
2 | 2.4272 | 0.131 i |
3 | -2.732 + 12.0437 i | 0.0946 - 0.053 i |
4 | -2.732 - 12.0437 i | 0.0946 + 0.053 i |
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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 | 1.4464 | 0.0772 |
![Singularities of quadratic [2, 2, 3] approximant](singsq2.gif?525988) |
2 | 2.3971 | 0.135 i |
3 | 3.7586 + 4.2208 i | 0.0537 - 0.119 i |
4 | 3.7586 - 4.2208 i | 0.0537 + 0.119 i |
5 | -6.5362 | 0.175 |
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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 | 1.2729 + 0.2991 i | 0.00862 - 0.0123 i |
![Singularities of quadratic [2, 3, 3] approximant](singsq3.gif?477099) |
2 | 1.2729 - 0.2991 i | 0.00862 + 0.0123 i |
3 | 1.547 | 0.0153 |
4 | 3.25 | 1.78 i |
5 | -9.3748 + 4.008 i | 0.269 - 0.105 i |
6 | -9.3748 - 4.008 i | 0.269 + 0.105 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 | 1.3534 + 0.3682 i | 0.0158 - 0.00653 i |
![Singularities of quadratic [3, 3, 3] approximant](singsq4.gif?376677) |
2 | 1.3534 - 0.3682 i | 0.0158 + 0.00653 i |
3 | 2.4158 | 0.171 |
4 | -3.2775 + 0.1918 i | 0.00483 + 0.00434 i |
5 | -3.2775 - 0.1918 i | 0.00483 - 0.00434 i |
6 | 5.796 | 0.0971 i |
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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 | 0.381 | 0.0000158 |
![Singularities of quadratic [3, 3, 4] approximant](singsq5.gif?882312) |
2 | 0.381 | 0.0000158 i |
3 | 1.3645 + 0.4434 i | 0.0114 + 0.00423 i |
4 | 1.3645 - 0.4434 i | 0.0114 - 0.00423 i |
5 | 3.7939 | 0.209 |
6 | -0.657 + 7.9333 i | 0.0162 - 0.0456 i |
7 | -0.657 - 7.9333 i | 0.0162 + 0.0456 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 | 1.4201 + 0.4445 i | 0.0139 + 0.0047 i |
![Singularities of quadratic [3, 4, 4] approximant](singsq6.gif?294673) |
2 | 1.4201 - 0.4445 i | 0.0139 - 0.0047 i |
3 | 1.0054 + 1.8291 i | 0.00341 + 0.00265 i |
4 | 1.0054 - 1.8291 i | 0.00341 - 0.00265 i |
5 | 0.8415 + 2.0518 i | 0.00326 - 0.00283 i |
6 | 0.8415 - 2.0518 i | 0.00326 + 0.00283 i |
7 | -5.5243 + 0.9471 i | 0.0128 + 0.00657 i |
8 | -5.5243 - 0.9471 i | 0.0128 - 0.00657 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 | 1.4627 + 0.3492 i | 0.0284 - 0.0196 i |
![Singularities of quadratic [4, 4, 4] approximant](singsq7.gif?403556) |
2 | 1.4627 - 0.3492 i | 0.0284 + 0.0196 i |
3 | 1.6751 + 1.3144 i | 0.002 - 0.0215 i |
4 | 1.6751 - 1.3144 i | 0.002 + 0.0215 i |
5 | 1.6947 + 1.9192 i | 0.0191 + 0.00744 i |
6 | 1.6947 - 1.9192 i | 0.0191 - 0.00744 i |
7 | 2.5361 + 7.8391 i | 0.0284 - 0.0251 i |
8 | 2.5361 - 7.8391 i | 0.0284 + 0.0251 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 | 1.4415 + 0.4181 i | 0.0193 + 0.00157 i |
![Singularities of quadratic [4, 4, 5] approximant](singsq8.gif?126167) |
2 | 1.4415 - 0.4181 i | 0.0193 - 0.00157 i |
3 | 1.469 + 1.5851 i | 0.0118 - 0.00139 i |
4 | 1.469 - 1.5851 i | 0.0118 + 0.00139 i |
5 | 1.5504 + 2.0885 i | 0.00269 - 0.0141 i |
6 | 1.5504 - 2.0885 i | 0.00269 + 0.0141 i |
7 | -9.2828 | 0.0831 |
8 | -21.3051 + 7.417 i | 0.597 + 0.815 i |
9 | -21.3051 - 7.417 i | 0.597 - 0.815 i |
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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 | 1.4358 + 0.1077 i | 0.018 - 0.00905 i |
![Singularities of quadratic [4, 5, 5] approximant](singsq9.gif?831166) |
2 | 1.4358 - 0.1077 i | 0.018 + 0.00905 i |
3 | 1.6068 + 0.6107 i | 0.0191 - 0.0226 i |
4 | 1.6068 - 0.6107 i | 0.0191 + 0.0226 i |
5 | 1.5522 + 2.1789 i | 0.00768 - 0.0221 i |
6 | 1.5522 - 2.1789 i | 0.00768 + 0.0221 i |
7 | 1.4766 + 2.7414 i | 0.0247 + 0.0137 i |
8 | 1.4766 - 2.7414 i | 0.0247 - 0.0137 i |
9 | -10.3097 + 3.5996 i | 0.343 - 0.0356 i |
10 | -10.3097 - 3.5996 i | 0.343 + 0.0356 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 | 1.4964 + 0.2976 i | 0.0207 - 0.0629 i |
![Singularities of quadratic [5, 5, 5] approximant](singsq10.gif?350802) |
2 | 1.4964 - 0.2976 i | 0.0207 + 0.0629 i |
3 | 1.9055 + 0.8803 i | 0.0418 + 0.0268 i |
4 | 1.9055 - 0.8803 i | 0.0418 - 0.0268 i |
5 | 1.8017 + 2.1756 i | 0.0275 - 0.013 i |
6 | 1.8017 - 2.1756 i | 0.0275 + 0.013 i |
7 | 1.9859 + 3.6028 i | 0.0044 + 0.0325 i |
8 | 1.9859 - 3.6028 i | 0.0044 - 0.0325 i |
9 | -4.6853 + 0.163 i | 0.00949 + 0.00919 i |
10 | -4.6853 - 0.163 i | 0.00949 - 0.00919 i |
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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 | 1.4876 + 0.3085 i | 0.0233 - 0.0496 i |
![Singularities of quadratic [5, 5, 6] approximant](singsq11.gif?620471) |
2 | 1.4876 - 0.3085 i | 0.0233 + 0.0496 i |
3 | 1.8796 + 1.0024 i | 0.0231 + 0.0345 i |
4 | 1.8796 - 1.0024 i | 0.0231 - 0.0345 i |
5 | 1.8801 + 2.3154 i | 0.0293 + 0.00538 i |
6 | 1.8801 - 2.3154 i | 0.0293 - 0.00538 i |
7 | 1.8634 + 4.4229 i | 0.0214 - 0.0309 i |
8 | 1.8634 - 4.4229 i | 0.0214 + 0.0309 i |
9 | -9.2236 + 2.3426 i | 0.516 + 1.18 i |
10 | -9.2236 - 2.3426 i | 0.516 - 1.18 i |
11 | -15.9494 | 0.236 |
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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 | 1.4851 + 0.3043 i | 0.0157 - 0.0484 i |
![Singularities of quadratic [5, 6, 6] approximant](singsq12.gif?901504) |
2 | 1.4851 - 0.3043 i | 0.0157 + 0.0484 i |
3 | 1.9151 + 1.0461 i | 0.00741 + 0.0417 i |
4 | 1.9151 - 1.0461 i | 0.00741 - 0.0417 i |
5 | 1.7312 + 2.6282 i | 0.00402 + 0.0276 i |
6 | 1.7312 - 2.6282 i | 0.00402 - 0.0276 i |
7 | -3.1516 + 1.71 i | 0.000384 + 0.00169 i |
8 | -3.1516 - 1.71 i | 0.000384 - 0.00169 i |
9 | -3.3377 + 1.6603 i | 0.00174 - 0.000263 i |
10 | -3.3377 - 1.6603 i | 0.00174 + 0.000263 i |
11 | 3.7252 + 2.7096 i | 0.135 + 0.102 i |
12 | 3.7252 - 2.7096 i | 0.135 - 0.102 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 | 1.5016 + 0.3139 i | 0.0532 - 0.0539 i |
![Singularities of quadratic [6, 6, 6] approximant](singsq13.gif?441266) |
2 | 1.5016 - 0.3139 i | 0.0532 + 0.0539 i |
3 | 1.8306 + 0.8659 i | 0.0337 - 0.00116 i |
4 | 1.8306 - 0.8659 i | 0.0337 + 0.00116 i |
5 | 1.649 + 1.8667 i | 0.00424 - 0.01 i |
6 | 1.649 - 1.8667 i | 0.00424 + 0.01 i |
7 | 0.822 + 3.5013 i | 0.00402 + 0.00476 i |
8 | 0.822 - 3.5013 i | 0.00402 - 0.00476 i |
9 | -6.1426 + 2.1283 i | 0.0102 + 0.00144 i |
10 | -6.1426 - 2.1283 i | 0.0102 - 0.00144 i |
11 | -0.9194 + 9.9324 i | 0.011 - 0.00232 i |
12 | -0.9194 - 9.9324 i | 0.011 + 0.00232 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 | 1.501 + 0.3068 i | 0.0278 - 0.0603 i |
![Singularities of quadratic [6, 6, 7] approximant](singsq14.gif?801824) |
2 | 1.501 - 0.3068 i | 0.0278 + 0.0603 i |
3 | 1.814 + 0.0616 i | 0.154 + 0.208 i |
4 | 1.814 - 0.0616 i | 0.154 - 0.208 i |
5 | 1.9027 + 1.0782 i | 0.0146 + 0.0397 i |
6 | 1.9027 - 1.0782 i | 0.0146 - 0.0397 i |
7 | 1.8998 + 2.3339 i | 0.0258 + 0.012 i |
8 | 1.8998 - 2.3339 i | 0.0258 - 0.012 i |
9 | 1.529 + 5.0743 i | 0.0308 - 0.0179 i |
10 | 1.529 - 5.0743 i | 0.0308 + 0.0179 i |
11 | -6.8288 + 6.5917 i | 0.117 - 0.0902 i |
12 | -6.8288 - 6.5917 i | 0.117 + 0.0902 i |
13 | -15.7944 | 0.193 |
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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 | 1.504 + 0.3198 i | 0.0568 - 0.0393 i |
![Singularities of quadratic [6, 7, 7] approximant](singsq15.gif?306698) |
2 | 1.504 - 0.3198 i | 0.0568 + 0.0393 i |
3 | 1.9598 + 0.7135 i | 0.0581 - 0.0602 i |
4 | 1.9598 - 0.7135 i | 0.0581 + 0.0602 i |
5 | 2.2441 | 0.0989 |
6 | 1.9699 + 1.5926 i | 0.0225 + 0.0312 i |
7 | 1.9699 - 1.5926 i | 0.0225 - 0.0312 i |
8 | 1.675 + 3.2409 i | 0.0277 - 0.00306 i |
9 | 1.675 - 3.2409 i | 0.0277 + 0.00306 i |
10 | -5.0831 + 4.1086 i | 0.0098 + 0.0367 i |
11 | -5.0831 - 4.1086 i | 0.0098 - 0.0367 i |
12 | -5.5154 + 6.4195 i | 0.032 - 0.0213 i |
13 | -5.5154 - 6.4195 i | 0.032 + 0.0213 i |
14 | 10.6968 | 0.0894 i |
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