Singularities of Møller-Plesset series: example "a86"

Molecule Cl-. Basis 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, 1, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
10.9619 + 2.3704 i
33.8 - 27.2 i
Singularities of quadratic [1, 1, 0] approximant
2
10.9619 - 2.3704 i
33.8 + 27.2 i
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Table 2. 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
-0.0621 + 0.e-4 i
0.000141 + 0.000141 i
Singularities of quadratic [1, 1, 1] approximant
2
-0.0621 - 0.e-4 i
0.000141 - 0.000141 i
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Table 3. 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
1.9122
0.13
Singularities of quadratic [2, 1, 1] approximant
2
3.6865
0.217 i
3
-8.5254
0.456
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Table 4. 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
2.6706 + 1.1672 i
0.281 - 0.0703 i
Singularities of quadratic [2, 2, 1] approximant
2
2.6706 - 1.1672 i
0.281 + 0.0703 i
3
0.5037 + 5.0135 i
0.378 - 0.238 i
4
0.5037 - 5.0135 i
0.378 + 0.238 i
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Table 5. 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
1.6279
0.0159
Singularities of quadratic [2, 2, 2] approximant
2
0.6864 + 1.9329 i
0.018 - 0.00288 i
3
0.6864 - 1.9329 i
0.018 + 0.00288 i
4
-3.6741
0.022
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Table 6. 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
2.1692
0.0782
Singularities of quadratic [3, 2, 2] approximant
2
1.3101 + 2.5604 i
0.092 - 0.0222 i
3
1.3101 - 2.5604 i
0.092 + 0.0222 i
4
-9.4876
0.132
5
37.7088
0.707 i
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Table 7. 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
2.4139
0.149
Singularities of quadratic [3, 3, 2] approximant
2
1.219 + 3.1114 i
0.139 + 0.0568 i
3
1.219 - 3.1114 i
0.139 - 0.0568 i
4
-3.9305
0.0456
5
-5.4053
0.0597 i
6
9.4008
3.15 i
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Table 8. 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
2.901
0.239
Singularities of quadratic [3, 3, 3] approximant
2
0.8961 + 2.9025 i
0.0789 + 0.0457 i
3
0.8961 - 2.9025 i
0.0789 - 0.0457 i
4
3.0508
0.419 i
5
-5.3235
0.0587
6
105.8513
0.139
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Table 9. 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
2.8305
1.97
Singularities of quadratic [4, 3, 3] approximant
2
0.3169 + 3.5227 i
0.178 - 0.0591 i
3
0.3169 - 3.5227 i
0.178 + 0.0591 i
4
2.3874 + 3.1655 i
0.123 + 0.163 i
5
2.3874 - 3.1655 i
0.123 - 0.163 i
6
11.5262
1.07 i
7
-61.2549
48.8
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Table 10. 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
0.2006
8.12e-7
Singularities of quadratic [4, 4, 3] approximant
2
0.2006
8.12e-7 i
3
2.2548
0.0411
4
0.4674 + 2.9645 i
0.00291 - 0.0471 i
5
0.4674 - 2.9645 i
0.00291 + 0.0471 i
6
3.2181
0.26 i
7
-9.8149 + 5.6703 i
0.333 - 0.155 i
8
-9.8149 - 5.6703 i
0.333 + 0.155 i
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Table 11. 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
2.9561
5.04
Singularities of quadratic [4, 4, 4] approximant
2
1.6854 + 3.0949 i
0.022 - 0.234 i
3
1.6854 - 3.0949 i
0.022 + 0.234 i
4
-0.3829 + 3.9821 i
0.0491 + 0.0835 i
5
-0.3829 - 3.9821 i
0.0491 - 0.0835 i
6
4.3125
0.818 i
7
-3.4443 + 4.4027 i
0.0828 + 0.014 i
8
-3.4443 - 4.4027 i
0.0828 - 0.014 i
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Table 12. 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
3.1343
17.
Singularities of quadratic [5, 4, 4] approximant
2
1.9774 + 2.9284 i
0.0803 + 0.139 i
3
1.9774 - 2.9284 i
0.0803 - 0.139 i
4
-0.018 + 3.8346 i
0.0879 + 0.0813 i
5
-0.018 - 3.8346 i
0.0879 - 0.0813 i
6
4.4929
0.41 i
7
-4.7664 + 2.7536 i
0.0272 + 0.0441 i
8
-4.7664 - 2.7536 i
0.0272 - 0.0441 i
9
-5.6419
0.0372
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Table 13. 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
2.8971 + 0.4374 i
0.463 - 0.58 i
Singularities of quadratic [5, 5, 4] approximant
2
2.8971 - 0.4374 i
0.463 + 0.58 i
3
1.1838 + 3.4857 i
0.407 + 0.166 i
4
1.1838 - 3.4857 i
0.407 - 0.166 i
5
4.777
1.85
6
-1.0819 + 4.9975 i
0.112 - 0.198 i
7
-1.0819 - 4.9975 i
0.112 + 0.198 i
8
-3.471 + 4.7312 i
0.194 + 0.133 i
9
-3.471 - 4.7312 i
0.194 - 0.133 i
10
9.7021
2.46 i
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Table 14. 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
2.9306 + 0.3958 i
0.331 - 0.858 i
Singularities of quadratic [5, 5, 5] approximant
2
2.9306 - 0.3958 i
0.331 + 0.858 i
3
1.0268 + 3.508 i
0.172 + 0.26 i
4
1.0268 - 3.508 i
0.172 - 0.26 i
5
-2.1448 + 4.5207 i
0.156 + 0.157 i
6
-2.1448 - 4.5207 i
0.156 - 0.157 i
7
6.1784
5.52
8
-3.0717 + 7.4103 i
0.266 + 0.132 i
9
-3.0717 - 7.4103 i
0.266 - 0.132 i
10
53.0274
0.17 i
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Table 15. 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
-1.1322
0.0000206
Singularities of quadratic [6, 5, 5] approximant
2
-1.1322
0.0000206 i
3
2.5857 + 0.1384 i
0.0774 - 0.0365 i
4
2.5857 - 0.1384 i
0.0774 + 0.0365 i
5
0.741 + 3.6378 i
0.0835 - 0.0617 i
6
0.741 - 3.6378 i
0.0835 + 0.0617 i
7
4.3615
0.356
8
-2.6709 + 3.5838 i
0.0215 - 0.0186 i
9
-2.6709 - 3.5838 i
0.0215 + 0.0186 i
10
0.2814 + 4.4798 i
0.0941 - 0.0597 i
11
0.2814 - 4.4798 i
0.0941 + 0.0597 i
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Table 16. 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
-1.1184
0.0000179
Singularities of quadratic [6, 6, 5] approximant
2
-1.1184
0.0000179 i
3
2.5699 + 0.1318 i
0.0693 - 0.0347 i
4
2.5699 - 0.1318 i
0.0693 + 0.0347 i
5
0.7188 + 3.6387 i
0.0789 - 0.0537 i
6
0.7188 - 3.6387 i
0.0789 + 0.0537 i
7
4.3739
0.351
8
-2.6607 + 3.5625 i
0.0205 - 0.0179 i
9
-2.6607 - 3.5625 i
0.0205 + 0.0179 i
10
0.2873 + 4.451 i
0.085 - 0.06 i
11
0.2873 - 4.451 i
0.085 + 0.06 i
12
16791.9716
15.5 i
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Table 17. 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
-0.8283
4.27e-6
Singularities of quadratic [6, 6, 6] approximant
2
-0.8283
4.27e-6 i
3
2.5505 + 0.125 i
0.06 - 0.0316 i
4
2.5505 - 0.125 i
0.06 + 0.0316 i
5
0.6854 + 3.6565 i
0.0821 - 0.0405 i
6
0.6854 - 3.6565 i
0.0821 + 0.0405 i
7
-2.6736 + 3.5533 i
0.0217 - 0.0168 i
8
-2.6736 - 3.5533 i
0.0217 + 0.0168 i
9
4.4609
0.355
10
0.2062 + 4.473 i
0.0813 - 0.044 i
11
0.2062 - 4.473 i
0.0813 + 0.044 i
12
-103.3353
318.
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Table 18. 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
0.6376 + 0.e-5 i
0.0000261 - 0.0000261 i
Singularities of quadratic [7, 6, 6] approximant
2
0.6376 - 0.e-5 i
0.0000261 + 0.0000261 i
3
-1.1349
0.0000202
4
-1.1349
0.0000202 i
5
2.5905 + 0.1401 i
0.0801 - 0.0372 i
6
2.5905 - 0.1401 i
0.0801 + 0.0372 i
7
0.741 + 3.6344 i
0.0822 - 0.0625 i
8
0.741 - 3.6344 i
0.0822 + 0.0625 i
9
4.3605
0.357
10
-2.6679 + 3.583 i
0.0214 - 0.0187 i
11
-2.6679 - 3.583 i
0.0214 + 0.0187 i
12
0.2859 + 4.4812 i
0.0942 - 0.0611 i
13
0.2859 - 4.4812 i
0.0942 + 0.0611 i
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Table 19. 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
-0.9851 + 1.3201 i
5.82e-6 - 0.000176 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.9851 - 1.3201 i
5.82e-6 + 0.000176 i
3
-0.985 + 1.3204 i
0.000176 + 5.84e-6 i
4
-0.985 - 1.3204 i
0.000176 - 5.84e-6 i
5
2.5027 + 0.104 i
0.0454 - 0.0286 i
6
2.5027 - 0.104 i
0.0454 + 0.0286 i
7
0.5451 + 3.5076 i
0.0306 - 0.0271 i
8
0.5451 - 3.5076 i
0.0306 + 0.0271 i
9
0.3343 + 4.1614 i
0.018 - 0.0538 i
10
0.3343 - 4.1614 i
0.018 + 0.0538 i
11
-2.5424 + 3.3612 i
0.0122 - 0.012 i
12
-2.5424 - 3.3612 i
0.0122 + 0.012 i
13
4.4053
0.319
14
210.2692
1.9 i
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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.1966
1.15e-10 - 1.15e-10 i
Singularities of quadratic [7, 7, 7] approximant
2
0.1966
1.15e-10 + 1.15e-10 i
3
-1.2397 + 1.2729 i
0.0000317 + 0.0000429 i
4
-1.2397 - 1.2729 i
0.0000317 - 0.0000429 i
5
-1.2417 + 1.2726 i
0.000043 - 0.0000316 i
6
-1.2417 - 1.2726 i
0.000043 + 0.0000316 i
7
2.4189 + 0.1023 i
0.0192 - 0.0106 i
8
2.4189 - 0.1023 i
0.0192 + 0.0106 i
9
-2.0721 + 3.4177 i
0.00703 + 0.00879 i
10
-2.0721 - 3.4177 i
0.00703 - 0.00879 i
11
1.2089 + 4.016 i
1.11 - 3.61 i
12
1.2089 - 4.016 i
1.11 + 3.61 i
13
4.2744 + 3.6015 i
0.193 - 0.304 i
14
4.2744 - 3.6015 i
0.193 + 0.304 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.