Singularities of Møller-Plesset series: example "o7"

Molecule CH2. Basis cc-pVTZ-(f/d). 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
1.5895
0.403
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
1.511
0.365
Singularities of quadratic [1, 1, 0] approximant
2
2416.859
14.6 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
2.1945
1.97
Singularities of quadratic [1, 1, 1] approximant
2
-4.8612
0.599
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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
-1.6556
0.246
Singularities of quadratic [2, 1, 1] approximant
2
1.7987
0.698
3
-1.9982
0.232 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
1.3217
0.126
Singularities of quadratic [2, 2, 1] approximant
2
-2.5341 + 2.2023 i
0.162 + 0.0427 i
3
-2.5341 - 2.2023 i
0.162 - 0.0427 i
4
4.7752
1.19 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
1.205 + 0.2792 i
0.0368 - 0.0303 i
Singularities of quadratic [2, 2, 2] approximant
2
1.205 - 0.2792 i
0.0368 + 0.0303 i
3
1.8615
0.103
4
-2.3641
0.0995
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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
1.156
0.0555
Singularities of quadratic [3, 2, 2] approximant
2
-1.9784 + 1.169 i
0.0409 + 0.022 i
3
-1.9784 - 1.169 i
0.0409 - 0.022 i
4
2.5849
0.236 i
5
-10.9544
0.221
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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
1.0421 + 0.1771 i
0.00613 - 0.0138 i
Singularities of quadratic [3, 3, 2] approximant
2
1.0421 - 0.1771 i
0.00613 + 0.0138 i
3
1.1399
0.0124
4
-2.5123
0.117
5
18.3276
1.5 i
6
-22.3891
0.675 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
1.1266
0.048
Singularities of quadratic [3, 3, 3] approximant
2
1.4467 + 1.4772 i
0.00735 + 0.0515 i
3
1.4467 - 1.4772 i
0.00735 - 0.0515 i
4
1.8193 + 1.3656 i
0.0572 - 0.024 i
5
1.8193 - 1.3656 i
0.0572 + 0.024 i
6
-2.6517
0.216
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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
1.0848
0.0258
Singularities of quadratic [4, 3, 3] approximant
2
-0.1243 + 1.1305 i
0.00292 + 0.000857 i
3
-0.1243 - 1.1305 i
0.00292 - 0.000857 i
4
-0.1429 + 1.141 i
0.000913 - 0.00295 i
5
-0.1429 - 1.141 i
0.000913 + 0.00295 i
6
-2.656
0.344
7
14.6598
0.643 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
-0.5019 + 0.0001 i
0.000117 + 0.000117 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.5019 - 0.0001 i
0.000117 - 0.000117 i
3
1.0984
0.0307
4
-2.3669
0.0433
5
3.7816
1.78 i
6
-4.794
2.32 i
7
3.1913 + 7.4017 i
0.105 - 0.351 i
8
3.1913 - 7.4017 i
0.105 + 0.351 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
-0.4766 + 0.3568 i
0.0000398 + 0.000138 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.4766 - 0.3568 i
0.0000398 - 0.000138 i
3
-0.4775 + 0.3566 i
0.000138 - 0.0000395 i
4
-0.4775 - 0.3566 i
0.000138 + 0.0000395 i
5
1.1065
0.0304
6
1.533
0.869 i
7
1.6936
0.161
8
-2.9744
4.56
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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
-0.8866 + 0.5063 i
0.0000416 + 0.00125 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.8866 - 0.5063 i
0.0000416 - 0.00125 i
3
-0.902 + 0.494 i
0.00124 - 9.11e-6 i
4
-0.902 - 0.494 i
0.00124 + 9.11e-6 i
5
1.1668 + 0.0734 i
0.0524 + 0.061 i
6
1.1668 - 0.0734 i
0.0524 - 0.061 i
7
1.3176
0.0492
8
-2.8203
7.66
9
13.8402
0.776 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
-0.8758 + 0.5019 i
0.0000355 + 0.00116 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.8758 - 0.5019 i
0.0000355 - 0.00116 i
3
-0.8904 + 0.4903 i
0.00115 - 6.61e-6 i
4
-0.8904 - 0.4903 i
0.00115 + 6.61e-6 i
5
1.159 + 0.0777 i
0.0452 + 0.0562 i
6
1.159 - 0.0777 i
0.0452 - 0.0562 i
7
1.2971
0.046
8
-2.857
18.
9
14.1672
0.729 i
10
-11742.8705
8.93 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
-0.8667 + 0.5797 i
0.000197 - 0.000399 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.8667 - 0.5797 i
0.000197 + 0.000399 i
3
1.0852 + 0.0563 i
0.0279 - 0.00101 i
4
1.0852 - 0.0563 i
0.0279 + 0.00101 i
5
-0.9101 + 0.5989 i
0.000397 + 0.000255 i
6
-0.9101 - 0.5989 i
0.000397 - 0.000255 i
7
-1.1149 + 0.0287 i
0.000214 + 0.000193 i
8
-1.1149 - 0.0287 i
0.000214 - 0.000193 i
9
1.1332
0.093
10
-2.2927
0.0246
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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
-0.8732 + 0.5813 i
0.000183 - 0.000408 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.8732 - 0.5813 i
0.000183 + 0.000408 i
3
1.0833 + 0.055 i
0.0272 - 0.00182 i
4
1.0833 - 0.055 i
0.0272 + 0.00182 i
5
-0.9207 + 0.6005 i
0.000403 + 0.000243 i
6
-0.9207 - 0.6005 i
0.000403 - 0.000243 i
7
1.1305
0.104
8
-1.171 + 0.0386 i
0.000256 + 0.00022 i
9
-1.171 - 0.0386 i
0.000256 - 0.00022 i
10
-2.249
0.0188
11
-291.0982
0.285 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
0.1417 + 0.e-5 i
1.46e-8 - 1.46e-8 i
Singularities of quadratic [6, 6, 5] approximant
2
0.1417 - 0.e-5 i
1.46e-8 + 1.46e-8 i
3
0.9859 + 0.0188 i
0.00333 - 0.00266 i
4
0.9859 - 0.0188 i
0.00333 + 0.00266 i
5
-0.923 + 0.5399 i
0.0007 + 0.000253 i
6
-0.923 - 0.5399 i
0.0007 - 0.000253 i
7
-0.9501 + 0.5773 i
0.000282 - 0.000771 i
8
-0.9501 - 0.5773 i
0.000282 + 0.000771 i
9
1.1476
3.57
10
-2.0916
0.0167
11
-31.2422
0.429 i
12
59.2998
0.773 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
0.0664 + 0.4183 i
3.25e-7 + 3.72e-6 i
Singularities of quadratic [6, 6, 6] approximant
2
0.0664 - 0.4183 i
3.25e-7 - 3.72e-6 i
3
0.0664 + 0.4183 i
3.72e-6 - 3.25e-7 i
4
0.0664 - 0.4183 i
3.72e-6 + 3.25e-7 i
5
-0.9008 + 0.5176 i
0.000391 + 0.0000883 i
6
-0.9008 - 0.5176 i
0.000391 - 0.0000883 i
7
-0.9424 + 0.5557 i
0.000106 - 0.000447 i
8
-0.9424 - 0.5557 i
0.000106 + 0.000447 i
9
1.1407 + 0.0956 i
0.00164 - 0.0647 i
10
1.1407 - 0.0956 i
0.00164 + 0.0647 i
11
1.2236
0.0361
12
-2.1716
0.0182
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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
0.1452 + 0.6517 i
0.0000641 + 0.0000687 i
Singularities of quadratic [7, 6, 6] approximant
2
0.1452 - 0.6517 i
0.0000641 - 0.0000687 i
3
0.1454 + 0.6519 i
0.0000688 - 0.0000641 i
4
0.1454 - 0.6519 i
0.0000688 + 0.0000641 i
5
-0.8995 + 0.5073 i
0.000116 + 0.000516 i
6
-0.8995 - 0.5073 i
0.000116 - 0.000516 i
7
-0.923 + 0.4769 i
0.000515 - 0.0000778 i
8
-0.923 - 0.4769 i
0.000515 + 0.0000778 i
9
1.1778 + 0.0674 i
0.0612 + 0.0568 i
10
1.1778 - 0.0674 i
0.0612 - 0.0568 i
11
1.3748
0.0554
12
-3.2759
0.402
13
8.0068
4.22 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
-0.5838
8.18e-7
Singularities of quadratic [7, 7, 6] approximant
2
-0.5852
8.18e-7 i
3
0.0975 + 0.8434 i
0.0000249 - 2.71e-6 i
4
0.0975 - 0.8434 i
0.0000249 + 2.71e-6 i
5
0.0904 + 0.8486 i
2.e-6 + 0.0000248 i
6
0.0904 - 0.8486 i
2.e-6 - 0.0000248 i
7
-0.8465 + 0.3674 i
0.0000145 - 0.0000129 i
8
-0.8465 - 0.3674 i
0.0000145 + 0.0000129 i
9
-0.8651 + 0.5241 i
0.0000287 + 0.0000151 i
10
-0.8651 - 0.5241 i
0.0000287 - 0.0000151 i
11
1.2731
0.096
12
1.8211
0.0735 i
13
4.2877 + 3.7926 i
0.0674 + 0.114 i
14
4.2877 - 3.7926 i
0.0674 - 0.114 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.