Singularities of Møller-Plesset series: example "a88"

Molecule Ne. 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
33.4218
13.8
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
0.2496
0.0115
Singularities of quadratic [1, 1, 0] approximant
2
0.2991
0.0126 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
-1.4264
0.239
Singularities of quadratic [1, 1, 1] approximant
2
-25.9117
0.369 i
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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
-0.7753 + 0.6567 i
0.0265 + 0.00763 i
Singularities of quadratic [2, 1, 1] approximant
2
-0.7753 - 0.6567 i
0.0265 - 0.00763 i
3
1.3546
0.0409
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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
-1.2088
0.172
Singularities of quadratic [2, 2, 1] approximant
2
-3.1197
0.254 i
3
3.8843
16.8
4
9.5498
1.13 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
-1.1091
0.0871
Singularities of quadratic [2, 2, 2] approximant
2
2.4781 + 1.1055 i
0.196 - 0.0351 i
3
2.4781 - 1.1055 i
0.196 + 0.0351 i
4
1045.7056
0.135
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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
0.2566
0.00023
Singularities of quadratic [3, 2, 2] approximant
2
0.2568
0.00023 i
3
-1.069
0.0554
4
2.1176
0.101
5
-4.1264
0.718 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
-1.0278
0.0697
Singularities of quadratic [3, 3, 2] approximant
2
-1.5038
0.0821 i
3
-1.779 + 0.9899 i
0.105 - 0.174 i
4
-1.779 - 0.9899 i
0.105 + 0.174 i
5
3.1821
1.89
6
15.1104
1.2 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
-0.9837
0.0463
Singularities of quadratic [3, 3, 3] approximant
2
-1.2729
0.0637 i
3
-1.992
0.449
4
-3.2806
0.16 i
5
3.1704 + 1.6133 i
0.432 + 0.146 i
6
3.1704 - 1.6133 i
0.432 - 0.146 i
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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
-0.9332
0.017
Singularities of quadratic [4, 3, 3] approximant
2
0.0081 + 1.4805 i
0.00727 + 0.000729 i
3
0.0081 - 1.4805 i
0.00727 - 0.000729 i
4
-1.57
0.0891 i
5
0.0832 + 1.5785 i
0.000446 - 0.00792 i
6
0.0832 - 1.5785 i
0.000446 + 0.00792 i
7
4.3422
1.06
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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
-0.9205
0.0237
Singularities of quadratic [4, 4, 3] approximant
2
-1.0904
0.0296 i
3
-1.5794
0.231
4
-1.5342 + 1.5981 i
0.0204 - 0.0914 i
5
-1.5342 - 1.5981 i
0.0204 + 0.0914 i
6
-2.5134
0.0958 i
7
3.6433
70.1
8
11.4282
0.855 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
-0.8827 + 0.0375 i
0.00537 + 0.00723 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.8827 - 0.0375 i
0.00537 - 0.00723 i
3
-0.9926
0.0118
4
3.1234
0.962
5
-4.0205
4.92 i
6
-3.123 + 3.6201 i
0.776 + 0.228 i
7
-3.123 - 3.6201 i
0.776 - 0.228 i
8
33.3157
0.987 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
-0.8748 + 0.0193 i
0.0108 + 0.015 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.8748 - 0.0193 i
0.0108 - 0.015 i
3
-1.0089
0.0223
4
-2.0545
0.545 i
5
-0.87 + 2.6186 i
0.0583 + 0.0216 i
6
-0.87 - 2.6186 i
0.0583 - 0.0216 i
7
3.8497
8.98
8
0.055 + 4.0544 i
0.0127 - 0.0928 i
9
0.055 - 4.0544 i
0.0127 + 0.0928 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
-0.8676 + 0.0231 i
0.007 + 0.00922 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.8676 - 0.0231 i
0.007 - 0.00922 i
3
-0.9898
0.0159
4
-2.1178
0.854 i
5
-0.9433 + 2.6073 i
0.0706 + 0.0274 i
6
-0.9433 - 2.6073 i
0.0706 - 0.0274 i
7
3.6681
380.
8
-0.482 + 4.3166 i
0.03 - 0.114 i
9
-0.482 - 4.3166 i
0.03 + 0.114 i
10
158.5786
7.11 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
-0.8328 + 0.0286 i
0.00161 + 0.00213 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.8328 - 0.0286 i
0.00161 - 0.00213 i
3
-0.9062
0.00352
4
1.838 + 0.0176 i
0.00687 - 0.00664 i
5
1.838 - 0.0176 i
0.00687 + 0.00664 i
6
-2.9288 + 1.9513 i
0.209 - 0.123 i
7
-2.9288 - 1.9513 i
0.209 + 0.123 i
8
0.1158 + 3.9463 i
0.15 + 0.0291 i
9
0.1158 - 3.9463 i
0.15 - 0.0291 i
10
4.0492
1.4
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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
-0.8285 + 0.0273 i
0.00144 + 0.00191 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.8285 - 0.0273 i
0.00144 - 0.00191 i
3
-0.8991
0.00314
4
-2.3518 + 1.5695 i
0.172 - 0.0131 i
5
-2.3518 - 1.5695 i
0.172 + 0.0131 i
6
3.3986 + 0.6183 i
0.227 + 0.537 i
7
3.3986 - 0.6183 i
0.227 - 0.537 i
8
0.3905 + 3.511 i
0.0604 + 0.0403 i
9
0.3905 - 3.511 i
0.0604 - 0.0403 i
10
2.6629 + 2.792 i
0.0244 + 0.112 i
11
2.6629 - 2.792 i
0.0244 - 0.112 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
0.3042 + 0.e-5 i
4.16e-7 - 4.16e-7 i
Singularities of quadratic [6, 6, 5] approximant
2
0.3042 - 0.e-5 i
4.16e-7 + 4.16e-7 i
3
-0.8214 + 0.0296 i
0.000967 + 0.00131 i
4
-0.8214 - 0.0296 i
0.000967 - 0.00131 i
5
-0.8811
0.00203
6
1.3072 + 2.8509 i
0.00688 + 0.0439 i
7
1.3072 - 2.8509 i
0.00688 - 0.0439 i
8
-3.4019
21. i
9
-2.0582 + 2.7664 i
0.058 + 0.104 i
10
-2.0582 - 2.7664 i
0.058 - 0.104 i
11
3.1277 + 2.7404 i
0.0749 - 0.0663 i
12
3.1277 - 2.7404 i
0.0749 + 0.0663 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
-0.8317 + 0.0028 i
0.0567 - 0.000758 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.8317 - 0.0028 i
0.0567 + 0.000758 i
3
-0.9203 + 0.0525 i
0.00213 + 0.00674 i
4
-0.9203 - 0.0525 i
0.00213 - 0.00674 i
5
-0.9534
0.00556
6
3.0625 + 0.6706 i
0.0664 - 0.729 i
7
3.0625 - 0.6706 i
0.0664 + 0.729 i
8
-3.7494
0.952 i
9
4.3954
0.725
10
-0.7078 + 4.549 i
0.0728 + 0.643 i
11
-0.7078 - 4.549 i
0.0728 - 0.643 i
12
-29.654
0.616
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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
-0.8283 + 0.0115 i
0.00555 + 0.00472 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.8283 - 0.0115 i
0.00555 - 0.00472 i
3
-0.9232 + 0.0419 i
0.000576 + 0.00923 i
4
-0.9232 - 0.0419 i
0.000576 - 0.00923 i
5
-0.9666
0.00755
6
3.0404 + 0.5651 i
0.397 + 0.555 i
7
3.0404 - 0.5651 i
0.397 - 0.555 i
8
4.3981
0.65
9
-0.3677 + 4.4135 i
0.229 + 0.361 i
10
-0.3677 - 4.4135 i
0.229 - 0.361 i
11
-5.028 + 0.5766 i
0.229 - 0.224 i
12
-5.028 - 0.5766 i
0.229 + 0.224 i
13
15.6351
8.71 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
-0.7805
0.000274
Singularities of quadratic [7, 7, 6] approximant
2
-0.791 + 0.0659 i
0.000284 - 0.0000789 i
3
-0.791 - 0.0659 i
0.000284 + 0.0000789 i
4
-0.7963 + 0.0527 i
0.0000316 + 0.00024 i
5
-0.7963 - 0.0527 i
0.0000316 - 0.00024 i
6
2.6224
0.34
7
2.7231
0.745 i
8
-2.7829
41.2 i
9
-1.5653 + 3.1093 i
0.0954 + 0.0497 i
10
-1.5653 - 3.1093 i
0.0954 - 0.0497 i
11
1.5654 + 3.5217 i
0.0426 + 0.0998 i
12
1.5654 - 3.5217 i
0.0426 - 0.0998 i
13
3.9151
2.07
14
11.2879
0.77 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.