Singularities of Møller-Plesset series: example "o5"

Molecule H2O. Basis cc-pVDZ(+). Structure ""

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Exampleso1o2o3o4o5o6o7o8o9
MoleculeNeNeF-HFH2OCH2CH2C2N2
Basiscc-pVDZcc-pVTZ-(f)cc-pVTZ-(f)cc-pVTZ-(f/d)cc-pVDZ(+)aug-cc-pVDZcc-pVTZ-(f/d)cc-pVDZ(+)cc-pVDZ

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Quadratic approximants

[n1n2n3] 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. zcc Location in the complex plane
1
10.7472
4.47
Singularities of quadratic [1, 0, 0] approximant
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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.5317
0.044
Singularities of quadratic [1, 1, 0] approximant
2
0.8794
0.0566 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. zcc Location in the complex plane
1
-2.0348
0.582
Singularities of quadratic [1, 1, 1] approximant
2
3.9655
24.3
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Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.4314
0.109
Singularities of quadratic [2, 1, 1] approximant
2
-1.2214 + 0.9806 i
0.0974 + 0.0246 i
3
-1.2214 - 0.9806 i
0.0974 - 0.0246 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. zcc Location in the complex plane
1
-1.2447
0.103
Singularities of quadratic [2, 2, 1] approximant
2
1.8738 + 1.8073 i
0.266 + 0.0843 i
3
1.8738 - 1.8073 i
0.266 - 0.0843 i
4
-7.9656
0.391 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. zcc Location in the complex plane
1
-1.0951 + 0.1958 i
0.0287 + 0.0337 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.0951 - 0.1958 i
0.0287 - 0.0337 i
3
-1.6277
0.0649
4
2.2167
0.487
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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.1388 + 0.1907 i
0.0428 + 0.0456 i
Singularities of quadratic [3, 2, 2] approximant
2
-1.1388 - 0.1907 i
0.0428 - 0.0456 i
3
-1.871
0.114
4
2.2211
0.517
5
-38.1604
1.51 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. zcc Location in the complex plane
1
-0.8881 + 0.1497 i
0.00114 + 0.00804 i
Singularities of quadratic [3, 3, 2] approximant
2
-0.8881 - 0.1497 i
0.00114 - 0.00804 i
3
-0.9012
0.00576
4
1.9972
0.21
5
9.3489
27.7 i
6
-57.9182
12.2 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. zcc Location in the complex plane
1
-0.2506 + 0.e-5 i
0.0000769 + 0.0000769 i
Singularities of quadratic [3, 3, 3] approximant
2
-0.2506 - 0.e-5 i
0.0000769 - 0.0000769 i
3
-1.0457
0.0397
4
-1.7698
0.128 i
5
2.1179
0.322
6
-2.892
3.92
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Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.0167
0.0467
Singularities of quadratic [4, 3, 3] approximant
2
-1.3454
0.0502 i
3
-1.7046 + 0.4529 i
0.0803 - 0.102 i
4
-1.7046 - 0.4529 i
0.0803 + 0.102 i
5
2.3388
1.08
6
-2.0343 + 3.0299 i
0.266 - 0.203 i
7
-2.0343 - 3.0299 i
0.266 + 0.203 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. zcc Location in the complex plane
1
-0.8822 + 0.0127 i
0.0028 + 0.00328 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.8822 - 0.0127 i
0.0028 - 0.00328 i
3
-0.966
0.00769
4
2.3418
1.
5
-3.7198
0.207 i
6
-1.0867 + 3.8914 i
0.0833 + 0.379 i
7
-1.0867 - 3.8914 i
0.0833 - 0.379 i
8
121.8585
4.93 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. zcc Location in the complex plane
1
-0.721 + 0.0022 i
0.00136 + 0.00137 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.721 - 0.0022 i
0.00136 - 0.00137 i
3
-0.9813
0.0152
4
2.3066
0.791
5
-0.7361 + 3.412 i
0.143 + 0.284 i
6
-0.7361 - 3.412 i
0.143 - 0.284 i
7
-4.0823 + 4.0349 i
0.319 + 0.296 i
8
-4.0823 - 4.0349 i
0.319 - 0.296 i
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Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.37
0.0000691
Singularities of quadratic [5, 4, 4] approximant
2
0.37
0.0000691 i
3
-0.7284 + 0.0024 i
0.000882 + 0.000886 i
4
-0.7284 - 0.0024 i
0.000882 - 0.000886 i
5
-0.9667
0.0113
6
2.4023
1.93
7
-3.4514
0.209 i
8
-1.6235 + 3.6419 i
0.234 - 0.432 i
9
-1.6235 - 3.6419 i
0.234 + 0.432 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. zcc Location in the complex plane
1
0.4271 + 0.e-5 i
0.000014 - 0.000014 i
Singularities of quadratic [5, 5, 4] approximant
2
0.4271 - 0.e-5 i
0.000014 + 0.000014 i
3
-0.9548
0.0136
4
-1.2509
0.0415 i
5
-1.4408
0.13
6
1.6461
0.0149
7
1.9576
0.0284 i
8
2.4586 + 2.8112 i
0.0958 + 0.143 i
9
2.4586 - 2.8112 i
0.0958 - 0.143 i
10
-4.4729
0.366 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. zcc Location in the complex plane
1
-0.2512 + 0.e-5 i
1.65e-6 + 1.65e-6 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.2512 - 0.e-5 i
1.65e-6 - 1.65e-6 i
3
-0.9506
0.0108
4
1.4332
0.00834
5
-1.4993
0.23 i
6
1.5314
0.00974 i
7
-1.9808
0.0693
8
2.8358 + 1.9931 i
0.197 - 0.029 i
9
2.8358 - 1.9931 i
0.197 + 0.029 i
10
-6.5942
3.05 i
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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.8217 + 0.0061 i
0.00149 + 0.00155 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.8217 - 0.0061 i
0.00149 - 0.00155 i
3
0.5786 + 0.7317 i
0.00484 + 0.0018 i
4
0.5786 - 0.7317 i
0.00484 - 0.0018 i
5
0.5796 + 0.7317 i
0.0018 - 0.00485 i
6
0.5796 - 0.7317 i
0.0018 + 0.00485 i
7
-0.9568
0.00793
8
2.3431
1.13
9
-3.0128
0.191 i
10
-1.9095 + 3.581 i
0.293 - 0.41 i
11
-1.9095 - 3.581 i
0.293 + 0.41 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. zcc Location in the complex plane
1
-0.8767 + 0.014 i
0.00219 + 0.0025 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.8767 - 0.014 i
0.00219 - 0.0025 i
3
0.4199 + 0.846 i
0.00146 + 0.00035 i
4
0.4199 - 0.846 i
0.00146 - 0.00035 i
5
0.4184 + 0.848 i
0.000354 - 0.00146 i
6
0.4184 - 0.848 i
0.000354 + 0.00146 i
7
-0.9593
0.00625
8
2.1319
0.202
9
-0.2568 + 3.5707 i
0.125 + 0.0989 i
10
-0.2568 - 3.5707 i
0.125 - 0.0989 i
11
-3.7238
0.196 i
12
13.4387
0.899 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. zcc Location in the complex plane
1
-0.8715 + 0.0132 i
0.00199 + 0.00224 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.8715 - 0.0132 i
0.00199 - 0.00224 i
3
0.4134 + 0.842 i
0.00127 + 0.000344 i
4
0.4134 - 0.842 i
0.00127 - 0.000344 i
5
0.4119 + 0.844 i
0.000348 - 0.00127 i
6
0.4119 - 0.844 i
0.000348 + 0.00127 i
7
-0.9566
0.00601
8
2.1162
0.179
9
-3.5836
0.188 i
10
-0.1901 + 3.6035 i
0.125 + 0.0903 i
11
-0.1901 - 3.6035 i
0.125 - 0.0903 i
12
10.28
0.859 i
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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.8961 + 0.0137 i
0.00454 + 0.00579 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.8961 - 0.0137 i
0.00454 - 0.00579 i
3
0.4209 + 0.8125 i
0.0015 - 0.000989 i
4
0.4209 - 0.8125 i
0.0015 + 0.000989 i
5
0.4223 + 0.8132 i
0.000992 + 0.0015 i
6
0.4223 - 0.8132 i
0.000992 - 0.0015 i
7
-0.9817
0.0107
8
1.9073 + 0.0317 i
0.216 - 0.145 i
9
1.9073 - 0.0317 i
0.216 + 0.145 i
10
2.5574
5.92
11
-3.8214
0.223 i
12
-1.624 + 3.7286 i
0.291 - 0.602 i
13
-1.624 - 3.7286 i
0.291 + 0.602 i
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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.3012 + 0.e-4 i
1.07e-6 + 1.07e-6 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.3012 - 0.e-4 i
1.07e-6 - 1.07e-6 i
3
-0.8472 + 0.0109 i
0.00112 + 0.00116 i
4
-0.8472 - 0.0109 i
0.00112 - 0.00116 i
5
0.4067 + 0.8395 i
0.00125 - 0.000415 i
6
0.4067 - 0.8395 i
0.00125 + 0.000415 i
7
0.4045 + 0.8405 i
0.000411 + 0.00125 i
8
0.4045 - 0.8405 i
0.000411 - 0.00125 i
9
-0.9433
0.00456
10
2.1225
0.199
11
0.0425 + 3.578 i
0.136 + 0.0405 i
12
0.0425 - 3.578 i
0.136 - 0.0405 i
13
-3.6361
0.186 i
14
9.7538
0.846 i
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Exampleso1o2o3o4o5o6o7o8o9
MoleculeNeNeF-HFH2OCH2CH2C2N2
Basiscc-pVDZcc-pVTZ-(f)cc-pVTZ-(f)cc-pVTZ-(f/d)cc-pVDZ(+)aug-cc-pVDZcc-pVTZ-(f/d)cc-pVDZ(+)cc-pVDZ

Plot of singularities Blank Molecule - icon for Allen-dataList of examples Blank Mathematica programs Blank Work in UMassD Blank Waste iconUnpublished reports

Designed by A. Sergeev.