Singularities of Møller-Plesset series: example "o6"

Molecule CH2. Basis aug-cc-pVDZ. Structure "theta=104.5`256"

Content


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
1.3682
0.317
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
1.6956
0.496
Singularities of quadratic [1, 1, 0] approximant
2
132.1892
4.38 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
1.9495
0.927
Singularities of quadratic [1, 1, 1] approximant
2
-14.4441
1.03
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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
-0.1168
0.000683
Singularities of quadratic [2, 1, 1] approximant
2
-0.117
0.000683 i
3
2.143
3.83
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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
1.3037
0.123
Singularities of quadratic [2, 2, 1] approximant
2
3.5333
0.555 i
3
-3.774 + 3.7843 i
0.215 + 0.0478 i
4
-3.774 - 3.7843 i
0.215 - 0.0478 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
0.606 + 0.0029 i
0.000649 - 0.000648 i
Singularities of quadratic [2, 2, 2] approximant
2
0.606 - 0.0029 i
0.000649 + 0.000648 i
3
1.0435
0.0133
4
-2.6308
0.0452
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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
1.1092 + 0.2492 i
0.0167 - 0.0183 i
Singularities of quadratic [3, 2, 2] approximant
2
1.1092 - 0.2492 i
0.0167 + 0.0183 i
3
1.4353
0.0316
4
-2.6768
0.0643
5
-11.2388
0.148 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
-0.4484 + 0.0002 i
0.0000812 + 0.0000811 i
Singularities of quadratic [3, 3, 2] approximant
2
-0.4484 - 0.0002 i
0.0000812 - 0.0000811 i
3
1.0953
0.0269
4
-2.1391
0.015
5
9.2922
0.9 i
6
-41.7268
0.302 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
-0.3458 + 0.e-4 i
0.0000215 + 0.0000215 i
Singularities of quadratic [3, 3, 3] approximant
2
-0.3458 - 0.e-4 i
0.0000215 - 0.0000215 i
3
1.0872
0.0244
4
-1.954
0.0106
5
-4.5661
0.042 i
6
-20.3596
0.169
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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
1.1409 + 0.1088 i
0.00645 - 0.0467 i
Singularities of quadratic [4, 3, 3] approximant
2
1.1409 - 0.1088 i
0.00645 + 0.0467 i
3
1.2694
0.0344
4
-1.8455 + 0.2343 i
0.00371 + 0.00545 i
5
-1.8455 - 0.2343 i
0.00371 - 0.00545 i
6
-2.3056
0.00746
7
8.051
3.19 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
1.1212
0.0421
Singularities of quadratic [4, 4, 3] approximant
2
1.7666
0.134 i
3
-1.8969 + 0.3188 i
0.00437 + 0.00605 i
4
-1.8969 - 0.3188 i
0.00437 - 0.00605 i
5
2.1383
46.4
6
-2.4751
0.00867
7
6.0147
114. i
8
-326.6589
2.78 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
-0.1546
6.89e-7
Singularities of quadratic [4, 4, 4] approximant
2
-0.1546
6.89e-7 i
3
0.9126
0.0529
4
0.9174
0.0333 i
5
1.071
0.0206
6
-1.8675
0.00987
7
-3.08
0.022 i
8
-7.3903
0.522
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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
-0.1337
1.06e-7
Singularities of quadratic [5, 4, 4] approximant
2
-0.1337
1.06e-7 i
3
0.2644 + 0.e-4 i
2.26e-6 - 2.26e-6 i
4
0.2644 - 0.e-4 i
2.26e-6 + 2.26e-6 i
5
1.1131
0.0416
6
-1.8221
0.00579
7
1.2383 + 3.8446 i
0.0446 - 0.044 i
8
1.2383 - 3.8446 i
0.0446 + 0.044 i
9
253.2882
4.56 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
-0.601 + 0.3795 i
0.0000188 + 0.000105 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.601 - 0.3795 i
0.0000188 - 0.000105 i
3
-0.6029 + 0.3803 i
0.000106 - 0.0000185 i
4
-0.6029 - 0.3803 i
0.000106 + 0.0000185 i
5
1.1606 + 0.1357 i
0.0318 - 0.0288 i
6
1.1606 - 0.1357 i
0.0318 + 0.0288 i
7
1.3799
0.0541
8
-1.8497
0.00484
9
5.2297
16.5 i
10
-795.5416
2.88 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
-0.8653 + 0.0776 i
0.0000338 + 0.0000702 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.8653 - 0.0776 i
0.0000338 - 0.0000702 i
3
-0.908 + 0.1095 i
0.0000912 - 0.0000209 i
4
-0.908 - 0.1095 i
0.0000912 + 0.0000209 i
5
1.1551
0.115
6
1.112 + 0.9766 i
0.00362 + 0.00905 i
7
1.112 - 0.9766 i
0.00362 - 0.00905 i
8
1.2599 + 0.9837 i
0.011 - 0.00535 i
9
1.2599 - 0.9837 i
0.011 + 0.00535 i
10
-1.8613
0.00291
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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
-0.6884 + 0.298 i
0.0000154 - 0.0000694 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.6884 - 0.298 i
0.0000154 + 0.0000694 i
3
-0.6922 + 0.2948 i
0.0000688 + 0.000016 i
4
-0.6922 - 0.2948 i
0.0000688 - 0.000016 i
5
0.7977 + 0.4317 i
0.00106 + 0.000812 i
6
0.7977 - 0.4317 i
0.00106 - 0.000812 i
7
0.8029 + 0.441 i
0.000853 - 0.00107 i
8
0.8029 - 0.441 i
0.000853 + 0.00107 i
9
1.1155
0.0549
10
-1.7904
0.00318
11
7.9057
4. 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
-0.5471 + 0.2361 i
3.31e-6 - 0.0000102 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.5471 - 0.2361 i
3.31e-6 + 0.0000102 i
3
-0.548 + 0.2348 i
0.0000101 + 3.33e-6 i
4
-0.548 - 0.2348 i
0.0000101 - 3.33e-6 i
5
0.718 + 0.8703 i
0.0000639 - 0.00105 i
6
0.718 - 0.8703 i
0.0000639 + 0.00105 i
7
1.161
0.234
8
0.7523 + 0.8912 i
0.00116 + 0.0000849 i
9
0.7523 - 0.8912 i
0.00116 - 0.0000849 i
10
-1.5983
0.0013
11
10.6894
10.6 i
12
-27.3287
2.29 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
0.5429
3.11e-6
Singularities of quadratic [6, 6, 6] approximant
2
0.5499
2.97e-6 i
3
0.6376
8.04e-6
4
-0.5898 + 0.266 i
8.5e-8 - 7.92e-6 i
5
-0.5898 - 0.266 i
8.5e-8 + 7.92e-6 i
6
-0.5926 + 0.2625 i
7.88e-6 + 1.87e-7 i
7
-0.5926 - 0.2625 i
7.88e-6 - 1.87e-7 i
8
0.6867
0.0000137 i
9
0.9492
0.000344
10
0.9439 + 0.2988 i
0.0000821 - 0.000669 i
11
0.9439 - 0.2988 i
0.0000821 + 0.000669 i
12
-1.5567
0.0008
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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
0.0316 + 0.592 i
3.24e-6 + 4.69e-6 i
Singularities of quadratic [7, 6, 6] approximant
2
0.0316 - 0.592 i
3.24e-6 - 4.69e-6 i
3
0.0321 + 0.5926 i
4.7e-6 - 3.24e-6 i
4
0.0321 - 0.5926 i
4.7e-6 + 3.24e-6 i
5
-0.7369 + 0.349 i
0.0000192 + 7.06e-6 i
6
-0.7369 - 0.349 i
0.0000192 - 7.06e-6 i
7
-0.7556 + 0.3739 i
7.32e-6 - 0.000021 i
8
-0.7556 - 0.3739 i
7.32e-6 + 0.000021 i
9
1.0615
0.00482
10
1.2421
0.0404 i
11
1.7137 + 0.6854 i
0.0331 - 0.0174 i
12
1.7137 - 0.6854 i
0.0331 + 0.0174 i
13
-3.0474
0.028
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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
-0.6268 + 0.0042 i
7.38e-7 + 7.26e-7 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.6268 - 0.0042 i
7.38e-7 - 7.26e-7 i
3
0.1154 + 0.7213 i
6.47e-6 - 7.64e-6 i
4
0.1154 - 0.7213 i
6.47e-6 + 7.64e-6 i
5
0.12 + 0.7257 i
7.84e-6 + 6.65e-6 i
6
0.12 - 0.7257 i
7.84e-6 - 6.65e-6 i
7
-0.604 + 0.4273 i
3.32e-7 - 3.69e-6 i
8
-0.604 - 0.4273 i
3.32e-7 + 3.69e-6 i
9
-0.6091 + 0.4549 i
4.13e-6 + 2.73e-7 i
10
-0.6091 - 0.4549 i
4.13e-6 - 2.73e-7 i
11
1.246
0.18
12
2.5693
28.1 i
13
-4.5436 + 2.5005 i
0.015 - 0.00917 i
14
-4.5436 - 2.5005 i
0.015 + 0.00917 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.