Singularities of Møller-Plesset series: example "a85"

Molecule F-. Basis aug-cc-pVDZ. Structure "mpn_Rfci"

Content


Examplesa1a2a8a16a22a30a38a44a45a51a62a69a75a83a84a85a86a87a88a90a91
MoleculeArBHBHBHBHBHBHBO+C2CN+N2HFHFHClHClF-Cl-Cl-NeOH-SH-
Basisaug-cc-pVDZcc-pVDZcc-pVTZcc-pVQZaug-cc-pVDZaug-cc-pVTZaug-cc-pVQZcc-pVDZcc-pVDZcc-pVDZcc-pVDZcc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZaug-cc-pVDZaug-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 [0, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [0, 0, 0] approximant
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Table 2. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [1, 0, 0] approximant
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Table 3. 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
-0.3984
0.0408
Singularities of quadratic [1, 1, 0] approximant
2
-0.7071
0.0544 i
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Table 4. 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
-0.8193
0.167
Singularities of quadratic [1, 1, 1] approximant
2
-8.0115
0.34 i
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Table 5. 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
-0.5577
0.0306
Singularities of quadratic [2, 1, 1] approximant
2
0.6173 + 1.2726 i
0.0586 + 0.012 i
3
0.6173 - 1.2726 i
0.0586 - 0.012 i
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Table 6. 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
-0.6193
0.0522
Singularities of quadratic [2, 2, 1] approximant
2
1.4706 + 1.4522 i
0.141 + 0.0225 i
3
1.4706 - 1.4522 i
0.141 - 0.0225 i
4
-8.5457
0.432 i
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Table 7. 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
-0.6128
0.0479
Singularities of quadratic [2, 2, 2] approximant
2
1.369 + 1.2226 i
0.102 - 0.00179 i
3
1.369 - 1.2226 i
0.102 + 0.00179 i
4
54.5903
5.25
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Table 8. 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
-0.6156
0.0501
Singularities of quadratic [3, 2, 2] approximant
2
1.5542 + 1.412 i
0.133 + 0.0237 i
3
1.5542 - 1.412 i
0.133 - 0.0237 i
4
1.7089 + 12.1401 i
0.732 + 0.28 i
5
1.7089 - 12.1401 i
0.732 - 0.28 i
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Table 9. 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
0.0902 + 0.e-4 i
0.0000406 - 0.0000406 i
Singularities of quadratic [3, 3, 2] approximant
2
0.0902 - 0.e-4 i
0.0000406 + 0.0000406 i
3
-0.6128
0.047
4
1.7076 + 1.4486 i
0.195 + 0.0108 i
5
1.7076 - 1.4486 i
0.195 - 0.0108 i
6
-8.5525
0.452 i
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Table 10. 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
-0.5518
0.00853
Singularities of quadratic [3, 3, 3] approximant
2
-0.5668 + 0.0285 i
0.00297 - 0.0108 i
3
-0.5668 - 0.0285 i
0.00297 + 0.0108 i
4
-2.025
0.217 i
5
2.7916
3.6
6
145.4765
0.301 i
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Table 11. 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
-0.4383
0.00218
Singularities of quadratic [4, 3, 3] approximant
2
-0.4403
0.00217 i
3
-0.5849
0.0198
4
-1.7662
0.311 i
5
2.7738
2.1
6
-5.0406
3.97
7
6.3926
2.21 i
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Table 12. 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
-0.5976
0.0339
Singularities of quadratic [4, 4, 3] approximant
2
-1.3389
451. i
3
1.3036 + 0.4299 i
0.00382 - 0.00847 i
4
1.3036 - 0.4299 i
0.00382 + 0.00847 i
5
1.5319 + 0.3677 i
0.0105 + 0.00078 i
6
1.5319 - 0.3677 i
0.0105 - 0.00078 i
7
-2.1942
0.142
8
-36.5721
299. i
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Table 13. 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
-0.5977
0.0341
Singularities of quadratic [4, 4, 4] approximant
2
1.2052
0.00496
3
-1.3157
139. i
4
1.3548 + 0.1861 i
0.00172 + 0.00656 i
5
1.3548 - 0.1861 i
0.00172 - 0.00656 i
6
-2.0531
0.131
7
2.2339
0.0515 i
8
-14.6185
2.85 i
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Table 14. 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
-0.6024
0.043
Singularities of quadratic [5, 4, 4] approximant
2
-0.9161
0.134 i
3
-1.0594
0.205
4
-0.1328 + 1.5366 i
0.0194 - 0.0263 i
5
-0.1328 - 1.5366 i
0.0194 + 0.0263 i
6
-0.2819 + 1.6515 i
0.0245 + 0.0221 i
7
-0.2819 - 1.6515 i
0.0245 - 0.0221 i
8
2.6345 + 0.5116 i
0.645 + 0.125 i
9
2.6345 - 0.5116 i
0.645 - 0.125 i
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Table 15. 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
-0.6066 + 0.034 i
0.00174 - 0.0453 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.6066 - 0.034 i
0.00174 + 0.0453 i
3
-0.6329
0.0302
4
-2.2039
0.254 i
5
2.0236 + 1.2868 i
0.176 + 0.0325 i
6
2.0236 - 1.2868 i
0.176 - 0.0325 i
7
1.6522 + 2.8634 i
0.197 - 0.147 i
8
1.6522 - 2.8634 i
0.197 + 0.147 i
9
-1.8542 + 4.3788 i
0.00362 - 1.52 i
10
-1.8542 - 4.3788 i
0.00362 + 1.52 i
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Table 16. 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
-0.6221 + 0.0277 i
0.00454 - 0.0905 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.6221 - 0.0277 i
0.00454 + 0.0905 i
3
-0.6697
0.052
4
2.1767 + 0.9368 i
0.314 - 0.15 i
5
2.1767 - 0.9368 i
0.314 + 0.15 i
6
-2.5685 + 1.3456 i
0.0331 - 0.223 i
7
-2.5685 - 1.3456 i
0.0331 + 0.223 i
8
3.5434
0.558
9
-1.5522 + 3.3303 i
0.575 - 0.687 i
10
-1.5522 - 3.3303 i
0.575 + 0.687 i
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Table 17. 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
-0.6183 + 0.0286 i
0.0123 - 0.0758 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.6183 - 0.0286 i
0.0123 + 0.0758 i
3
0.6427 + 0.e-4 i
0.000221 - 0.000221 i
4
0.6427 - 0.e-4 i
0.000221 + 0.000221 i
5
-0.6596
0.0453
6
1.6936 + 0.1268 i
0.0203 - 0.0131 i
7
1.6936 - 0.1268 i
0.0203 + 0.0131 i
8
-0.6092 + 2.3399 i
0.12 - 0.0055 i
9
-0.6092 - 2.3399 i
0.12 + 0.0055 i
10
-5.8909 + 1.4314 i
0.0501 - 0.0776 i
11
-5.8909 - 1.4314 i
0.0501 + 0.0776 i
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Table 18. 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
-0.6306 + 0.0209 i
0.0142 - 0.16 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.6306 - 0.0209 i
0.0142 + 0.16 i
3
-0.6932
0.0748
4
-1.3501
0.23 i
5
-1.708 + 0.1024 i
0.0835 + 0.146 i
6
-1.708 - 0.1024 i
0.0835 - 0.146 i
7
2.255 + 0.9399 i
0.413 - 0.328 i
8
2.255 - 0.9399 i
0.413 + 0.328 i
9
-0.5657 + 3.7241 i
0.397 - 0.394 i
10
-0.5657 - 3.7241 i
0.397 + 0.394 i
11
6.3858
0.742
12
9.7415
2.28 i
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Table 19. 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
-0.6087
0.0549
Singularities of quadratic [6, 6, 6] approximant
2
-0.6383 + 0.0511 i
0.151 + 0.0327 i
3
-0.6383 - 0.0511 i
0.151 - 0.0327 i
4
-0.6542 + 0.0394 i
0.034 - 0.05 i
5
-0.6542 - 0.0394 i
0.034 + 0.05 i
6
2.071 + 1.0524 i
0.208 - 0.0283 i
7
2.071 - 1.0524 i
0.208 + 0.0283 i
8
-3.2715 + 1.545 i
0.184 - 0.345 i
9
-3.2715 - 1.545 i
0.184 + 0.345 i
10
-4.5698
0.262 i
11
5.181
2.32
12
-11.0002
0.376
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Table 20. 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
-0.6389 + 0.0084 i
0.295 - 0.445 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.6389 - 0.0084 i
0.295 + 0.445 i
3
-0.7549 + 0.0224 i
0.195 + 0.00388 i
4
-0.7549 - 0.0224 i
0.195 - 0.00388 i
5
-0.8195
0.148
6
1.9836 + 1.024 i
0.153 - 0.0564 i
7
1.9836 - 1.024 i
0.153 + 0.0564 i
8
-1.203 + 2.8315 i
0.454 + 0.105 i
9
-1.203 - 2.8315 i
0.454 - 0.105 i
10
-3.1159
0.121 i
11
2.9446 + 1.7773 i
0.043 + 0.303 i
12
2.9446 - 1.7773 i
0.043 - 0.303 i
13
-5.4539
0.19
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Examplesa1a2a8a16a22a30a38a44a45a51a62a69a75a83a84a85a86a87a88a90a91
MoleculeArBHBHBHBHBHBHBO+C2CN+N2HFHFHClHClF-Cl-Cl-NeOH-SH-
Basisaug-cc-pVDZcc-pVDZcc-pVTZcc-pVQZaug-cc-pVDZaug-cc-pVTZaug-cc-pVQZcc-pVDZcc-pVDZcc-pVDZcc-pVDZcc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZaug-cc-pVDZaug-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.