Singularities of Møller-Plesset series: example "a91"

Molecule SH-. 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 [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
2.4002
0.766
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
1.5521
0.34
Singularities of quadratic [1, 1, 0] approximant
2
40.4586
1.73 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
-2.6469
0.361
Singularities of quadratic [1, 1, 1] approximant
2
4.1358
104.
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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
-1.8687
0.82
Singularities of quadratic [2, 1, 1] approximant
2
2.2299
0.861
3
-2.3614
0.65 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
-1.4895
0.136
Singularities of quadratic [2, 2, 1] approximant
2
-2.4668
0.158 i
3
2.7328
5.92
4
44.8059
1.53 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
-1.3388 + 0.3464 i
0.0268 + 0.0285 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.3388 - 0.3464 i
0.0268 - 0.0285 i
3
2.1102
0.393
4
-2.598
0.0661
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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
-1.2009
0.0414
Singularities of quadratic [3, 2, 2] approximant
2
-1.8215
0.0412 i
3
2.546
3.37
4
-2.3521 + 1.4337 i
0.0977 - 0.0583 i
5
-2.3521 - 1.4337 i
0.0977 + 0.0583 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
-1.1967
0.0334
Singularities of quadratic [3, 3, 2] approximant
2
-2.4039
0.0397 i
3
2.6685
11.9
4
-2.0207 + 2.04 i
0.0498 - 0.0638 i
5
-2.0207 - 2.04 i
0.0498 + 0.0638 i
6
272.1259
20.5 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
-0.5961
0.000424
Singularities of quadratic [3, 3, 3] approximant
2
-0.5979
0.000424 i
3
-1.0953
0.00921
4
2.4451 + 0.2497 i
0.359 - 0.047 i
5
2.4451 - 0.2497 i
0.359 + 0.047 i
6
5.488
2.04
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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
-0.1849 + 0.e-5 i
3.39e-6 + 3.39e-6 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.1849 - 0.e-5 i
3.39e-6 - 3.39e-6 i
3
-1.1273
0.0139
4
2.4642
0.67
5
0.3799 + 4.7768 i
0.17 + 0.0121 i
6
0.3799 - 4.7768 i
0.17 - 0.0121 i
7
4.9531
1.35 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
-1.1246
0.0146
Singularities of quadratic [4, 4, 3] approximant
2
-0.5896 + 1.5245 i
0.00125 - 0.00836 i
3
-0.5896 - 1.5245 i
0.00125 + 0.00836 i
4
-0.6911 + 1.5232 i
0.00817 + 0.00195 i
5
-0.6911 - 1.5232 i
0.00817 - 0.00195 i
6
3.0282
6.09
7
-6.2685
1.53 i
8
30.7528
424. 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
-0.7572 + 0.0012 i
0.000747 + 0.00075 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.7572 - 0.0012 i
0.000747 - 0.00075 i
3
-1.1197
0.0114
4
2.8174
948.
5
0.0406 + 3.0437 i
0.0548 + 0.0377 i
6
0.0406 - 3.0437 i
0.0548 - 0.0377 i
7
0.9531 + 3.6018 i
0.053 - 0.0915 i
8
0.9531 - 3.6018 i
0.053 + 0.0915 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
-0.1551
3.61e-7
Singularities of quadratic [5, 4, 4] approximant
2
-0.1551
3.61e-7 i
3
-1.0194
0.00716
4
-1.0617
0.00976 i
5
-1.1981
0.0465
6
2.4493
0.671
7
-0.008 + 4.1411 i
0.12 + 0.0161 i
8
-0.008 - 4.1411 i
0.12 - 0.0161 i
9
6.5688
0.774 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
-1.0187 + 0.0165 i
0.00313 + 0.00389 i
Singularities of quadratic [5, 5, 4] approximant
2
-1.0187 - 0.0165 i
0.00313 - 0.00389 i
3
-1.123
0.00758
4
1.8183 + 0.0071 i
0.342 - 0.291 i
5
1.8183 - 0.0071 i
0.342 + 0.291 i
6
2.9294
97.2
7
-1.3213 + 3.4726 i
0.0122 + 0.0975 i
8
-1.3213 - 3.4726 i
0.0122 - 0.0975 i
9
-8.4077
0.142 i
10
16.2366
2.72 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
-1.037 + 0.0182 i
0.00431 + 0.00588 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.037 - 0.0182 i
0.00431 - 0.00588 i
3
-1.1369
0.00917
4
1.5972 + 0.0108 i
0.0188 - 0.0182 i
5
1.5972 - 0.0108 i
0.0188 + 0.0182 i
6
2.6552
2.47
7
-1.1894 + 3.6481 i
0.0344 + 0.0902 i
8
-1.1894 - 3.6481 i
0.0344 - 0.0902 i
9
-7.7348
0.158 i
10
8.5318
0.855 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
-1.0367 + 0.0182 i
0.00428 + 0.00583 i
Singularities of quadratic [6, 5, 5] approximant
2
-1.0367 - 0.0182 i
0.00428 - 0.00583 i
3
-1.1366
0.00912
4
1.5956 + 0.0106 i
0.0188 - 0.0182 i
5
1.5956 - 0.0106 i
0.0188 + 0.0182 i
6
2.655
2.48
7
-1.1956 + 3.6536 i
0.0343 + 0.0909 i
8
-1.1956 - 3.6536 i
0.0343 - 0.0909 i
9
-7.594
0.155 i
10
8.5786
0.855 i
11
-660.0826
3.01
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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
-0.3471 + 0.e-5 i
4.49e-6 + 4.49e-6 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.3471 - 0.e-5 i
4.49e-6 - 4.49e-6 i
3
-1.0506 + 0.0129 i
0.0086 + 0.0158 i
4
-1.0506 - 0.0129 i
0.0086 - 0.0158 i
5
-1.1571
0.0139
6
1.6694 + 0.0047 i
0.146 - 0.14 i
7
1.6694 - 0.0047 i
0.146 + 0.14 i
8
2.8087
53.1
9
-1.1014 + 3.6197 i
0.0396 + 0.0927 i
10
-1.1014 - 3.6197 i
0.0396 - 0.0927 i
11
-13.041
0.241 i
12
17.3669
2.38 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
-1. + 0.0163 i
0.00168 + 0.00196 i
Singularities of quadratic [6, 6, 6] approximant
2
-1. - 0.0163 i
0.00168 - 0.00196 i
3
-1.0965
0.00459
4
1.8281
0.0341
5
1.9636
0.0411 i
6
-2.9786
0.082 i
7
1.0363 + 2.9981 i
0.0591 - 0.0125 i
8
1.0363 - 2.9981 i
0.0591 + 0.0125 i
9
0.5214 + 4.1245 i
0.00165 - 0.0671 i
10
0.5214 - 4.1245 i
0.00165 + 0.0671 i
11
-7.7237
0.293
12
8.8415
41.5
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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
-0.9949 + 0.0133 i
0.00191 + 0.00224 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.9949 - 0.0133 i
0.00191 - 0.00224 i
3
-1.0976
0.00529
4
-1.8269
22.9 i
5
-1.9681
0.0737
6
2.0022 + 0.4727 i
0.062 + 0.0158 i
7
2.0022 - 0.4727 i
0.062 - 0.0158 i
8
1.996 + 0.6334 i
0.0345 - 0.0569 i
9
1.996 - 0.6334 i
0.0345 + 0.0569 i
10
2.5215
69.8
11
-0.4675 + 3.9712 i
0.0974 + 0.0513 i
12
-0.4675 - 3.9712 i
0.0974 - 0.0513 i
13
10.1045
0.694 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
-0.9813 + 0.0158 i
0.00091 + 0.00101 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.9813 - 0.0158 i
0.00091 - 0.00101 i
3
-1.0735
0.00279
4
1.8313
0.0161
5
2.1572
0.0322 i
6
1.288 + 2.3385 i
0.0133 - 0.00201 i
7
1.288 - 2.3385 i
0.0133 + 0.00201 i
8
1.741 + 2.0587 i
0.00876 + 0.0147 i
9
1.741 - 2.0587 i
0.00876 - 0.0147 i
10
-1.3392 + 2.6091 i
0.0159 - 0.00786 i
11
-1.3392 - 2.6091 i
0.0159 + 0.00786 i
12
-3.2054
0.0661 i
13
-2.5721 + 2.4929 i
0.00589 + 0.0282 i
14
-2.5721 - 2.4929 i
0.00589 - 0.0282 i
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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.