Singularities of Møller-Plesset series: example "a8"
Molecule BH. Basis cc-pVTZ. 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 | 1.1098 | 0.163 |
![Singularities of quadratic [1, 0, 0] approximant](singsq1.gif?73005) |
Top of the page
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 | 1.4066 | 0.268 |
![Singularities of quadratic [1, 1, 0] approximant](singsq2.gif?341726) |
2 | 88.8986 | 2.13 i |
Top of the page
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 | 1.5231 | 0.379 |
![Singularities of quadratic [1, 1, 1] approximant](singsq3.gif?205890) |
2 | -23.1777 | 0.731 |
Top of the page
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 | 1.2212 | 0.11 |
![Singularities of quadratic [2, 1, 1] approximant](singsq4.gif?574607) |
2 | 1.9046 | 0.204 i |
3 | 3.3308 | 0.699 |
Top of the page
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 | 1.4404 | 0.273 |
![Singularities of quadratic [2, 2, 1] approximant](singsq5.gif?98302) |
2 | -7.4525 + 1.4664 i | 0.66 + 0.323 i |
3 | -7.4525 - 1.4664 i | 0.66 - 0.323 i |
4 | 28.4954 | 2.07 i |
Top of the page
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 | 1.6843 | 4.37 |
![Singularities of quadratic [2, 2, 2] approximant](singsq6.gif?41888) |
2 | 2.0332 | 1.29 i |
3 | 3.3141 | 24.7 |
4 | -7.0871 | 0.215 |
Top of the page
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 | 1.595 + 0.3265 i | 0.309 - 0.222 i |
![Singularities of quadratic [3, 2, 2] approximant](singsq7.gif?776044) |
2 | 1.595 - 0.3265 i | 0.309 + 0.222 i |
3 | 2.2951 | 0.641 |
4 | -5.7901 | 0.156 |
5 | -20.7909 | 0.186 i |
Top of the page
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 | 1.5733 + 0.3243 i | 0.269 - 0.237 i |
![Singularities of quadratic [3, 3, 2] approximant](singsq8.gif?497117) |
2 | 1.5733 - 0.3243 i | 0.269 + 0.237 i |
3 | 2.1857 | 0.493 |
4 | -6.0635 | 0.181 |
5 | -23.5914 | 0.215 i |
6 | 1245.2983 | 1.93 i |
Top of the page
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 | 1.5729 + 0.3239 i | 0.268 - 0.239 i |
![Singularities of quadratic [3, 3, 3] approximant](singsq9.gif?526806) |
2 | 1.5729 - 0.3239 i | 0.268 + 0.239 i |
3 | 2.1827 | 0.489 |
4 | -6.0709 | 0.184 |
5 | -21.9614 | 0.224 i |
6 | -418.9354 | 2.16 |
Top of the page
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 | 0.5746 + 0.e-4 i | 0.000897 - 0.000897 i |
![Singularities of quadratic [4, 3, 3] approximant](singsq10.gif?247963) |
2 | 0.5746 - 0.e-4 i | 0.000897 + 0.000897 i |
3 | 1.5505 + 0.2966 i | 0.128 - 0.332 i |
4 | 1.5505 - 0.2966 i | 0.128 + 0.332 i |
5 | 2.1218 | 0.39 |
6 | -5.8007 | 0.14 |
7 | -28.5096 | 0.192 i |
Top of the page
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 | 1.5902 + 0.366 i | 0.272 - 0.0715 i |
![Singularities of quadratic [4, 4, 3] approximant](singsq11.gif?736123) |
2 | 1.5902 - 0.366 i | 0.272 + 0.0715 i |
3 | 2.346 | 0.97 |
4 | -3.8027 | 0.026 |
5 | -4.9206 | 0.0275 i |
6 | -4.6869 + 7.0614 i | 0.0772 - 0.0877 i |
7 | -4.6869 - 7.0614 i | 0.0772 + 0.0877 i |
8 | 33.7868 | 5.52 i |
Top of the page
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 | 1.5912 + 0.3735 i | 0.27 - 0.029 i |
![Singularities of quadratic [4, 4, 4] approximant](singsq12.gif?970352) |
2 | 1.5912 - 0.3735 i | 0.27 + 0.029 i |
3 | 2.248 | 0.772 |
4 | -0.6592 + 2.4583 i | 0.00443 + 0.0109 i |
5 | -0.6592 - 2.4583 i | 0.00443 - 0.0109 i |
6 | -0.679 + 2.5177 i | 0.0113 - 0.00419 i |
7 | -0.679 - 2.5177 i | 0.0113 + 0.00419 i |
8 | -4.9634 | 0.0503 |
Top of the page
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 | 1.5787 + 0.2702 i | 0.0797 + 0.372 i |
![Singularities of quadratic [5, 4, 4] approximant](singsq13.gif?958065) |
2 | 1.5787 - 0.2702 i | 0.0797 - 0.372 i |
3 | 2.2061 + 0.727 i | 0.224 + 0.211 i |
4 | 2.2061 - 0.727 i | 0.224 - 0.211 i |
5 | -2.4712 + 0.0271 i | 0.00306 + 0.00308 i |
6 | -2.4712 - 0.0271 i | 0.00306 - 0.00308 i |
7 | 3.2013 | 0.561 |
8 | -4.9032 | 0.0325 |
9 | 16.4278 | 0.564 i |
Top of the page
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 | 1.3998 + 0.0329 i | 0.0394 - 0.0367 i |
![Singularities of quadratic [5, 5, 4] approximant](singsq14.gif?263093) |
2 | 1.3998 - 0.0329 i | 0.0394 + 0.0367 i |
3 | 1.6915 + 0.4481 i | 0.0679 + 0.341 i |
4 | 1.6915 - 0.4481 i | 0.0679 - 0.341 i |
5 | 2.1807 | 1.16 |
6 | -3.7952 + 0.0761 i | 0.102 + 0.0968 i |
7 | -3.7952 - 0.0761 i | 0.102 - 0.0968 i |
8 | -9.4334 | 1.08 |
9 | -17.5377 | 1.01 i |
10 | 34.1665 | 1.23 i |
Top of the page
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 | 1.4097 + 0.0365 i | 0.0416 - 0.038 i |
![Singularities of quadratic [5, 5, 5] approximant](singsq15.gif?532621) |
2 | 1.4097 - 0.0365 i | 0.0416 + 0.038 i |
3 | 1.6983 + 0.4525 i | 0.0356 + 0.345 i |
4 | 1.6983 - 0.4525 i | 0.0356 - 0.345 i |
5 | 2.1646 | 1.15 |
6 | -3.8295 + 0.0977 i | 0.0787 + 0.0736 i |
7 | -3.8295 - 0.0977 i | 0.0787 - 0.0736 i |
8 | -10.7667 + 2.519 i | 0.434 + 1.18 i |
9 | -10.7667 - 2.519 i | 0.434 - 1.18 i |
10 | 66.1385 | 0.427 i |
Top of the page
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 | 1.4258 + 0.0499 i | 0.038 - 0.0309 i |
![Singularities of quadratic [6, 5, 5] approximant](singsq16.gif?593840) |
2 | 1.4258 - 0.0499 i | 0.038 + 0.0309 i |
3 | 1.7378 + 0.4815 i | 0.17 - 0.267 i |
4 | 1.7378 - 0.4815 i | 0.17 + 0.267 i |
5 | 1.9955 | 0.743 |
6 | -3.3634 + 0.2346 i | 0.00327 + 0.00418 i |
7 | -3.3634 - 0.2346 i | 0.00327 - 0.00418 i |
8 | -4.2643 | 0.00754 |
9 | -1.0869 + 10.6297 i | 0.144 + 0.088 i |
10 | -1.0869 - 10.6297 i | 0.144 - 0.088 i |
11 | -13.2713 | 0.121 i |
Top of the page
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 | 1.4444 + 0.0692 i | 0.039 - 0.0286 i |
![Singularities of quadratic [6, 6, 5] approximant](singsq17.gif?29081) |
2 | 1.4444 - 0.0692 i | 0.039 + 0.0286 i |
3 | 1.7233 + 0.5483 i | 0.108 - 0.124 i |
4 | 1.7233 - 0.5483 i | 0.108 + 0.124 i |
5 | 1.8216 | 0.528 |
6 | 2.862 | 0.678 i |
7 | -3.6866 + 0.2224 i | 0.015 + 0.0146 i |
8 | -3.6866 - 0.2224 i | 0.015 - 0.0146 i |
9 | 3.7212 | 1.25 |
10 | -6.8583 | 0.086 |
11 | 18.0958 | 2.98 i |
12 | -27.5978 | 1.28 i |
Top of the page
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 | 0.3727 + 0.e-5 i | 1.44e-7 - 1.44e-7 i |
![Singularities of quadratic [6, 6, 6] approximant](singsq18.gif?106052) |
2 | 0.3727 - 0.e-5 i | 1.44e-7 + 1.44e-7 i |
3 | 1.1773 + 0.0065 i | 0.000619 - 0.0006 i |
4 | 1.1773 - 0.0065 i | 0.000619 + 0.0006 i |
5 | 1.4178 | 0.0127 |
6 | 1.6929 + 0.6798 i | 0.0406 - 0.00596 i |
7 | 1.6929 - 0.6798 i | 0.0406 + 0.00596 i |
8 | 2.8668 | 0.496 i |
9 | -3.44 | 0.00855 |
10 | -4.4027 | 0.0123 i |
11 | 4.9751 | 1.09 |
12 | -14.3294 | 0.222 |
Top of the page
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 | 1.3452 + 0.1314 i | 0.0405 + 0.0124 i |
![Singularities of quadratic [7, 6, 6] approximant](singsq19.gif?41837) |
2 | 1.3452 - 0.1314 i | 0.0405 - 0.0124 i |
3 | 1.3586 + 0.107 i | 0.0153 - 0.0388 i |
4 | 1.3586 - 0.107 i | 0.0153 + 0.0388 i |
5 | 1.6981 + 0.4596 i | 0.0258 - 0.341 i |
6 | 1.6981 - 0.4596 i | 0.0258 + 0.341 i |
7 | 2.0846 | 0.786 |
8 | -3.609 + 0.2265 i | 0.0104 + 0.0114 i |
9 | -3.609 - 0.2265 i | 0.0104 - 0.0114 i |
10 | -5.6194 | 0.0335 |
11 | -0.0007 + 13.8331 i | 0.235 + 0.203 i |
12 | -0.0007 - 13.8331 i | 0.235 - 0.203 i |
13 | -35.1917 | 1.33 i |
Top of the page
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 | 1.4624 | 0.0386 |
![Singularities of quadratic [7, 7, 6] approximant](singsq20.gif?383497) |
2 | 1.4272 + 0.3745 i | 0.0121 + 0.00503 i |
3 | 1.4272 - 0.3745 i | 0.0121 - 0.00503 i |
4 | 1.4101 + 0.439 i | 0.00685 - 0.0103 i |
5 | 1.4101 - 0.439 i | 0.00685 + 0.0103 i |
6 | 1.9162 + 0.6441 i | 0.0472 + 0.0992 i |
7 | 1.9162 - 0.6441 i | 0.0472 - 0.0992 i |
8 | -3.3868 | 0.00702 |
9 | -4.5445 | 0.00859 i |
10 | -5.7371 + 4.805 i | 0.00142 - 0.0249 i |
11 | -5.7371 - 4.805 i | 0.00142 + 0.0249 i |
12 | -4.0523 + 7.3331 i | 0.0313 - 0.0184 i |
13 | -4.0523 - 7.3331 i | 0.0313 + 0.0184 i |
14 | 14.9408 | 13.8 i |
Top of the page
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.