Singularities of Møller-Plesset series: example "a87"

Molecule Cl-. 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
3.2208
1.02
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
2.0569
0.445
Singularities of quadratic [1, 1, 0] approximant
2
50.988
2.21 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
-2.9903
0.36
Singularities of quadratic [1, 1, 1] approximant
2
7.5685
5.12
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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
-1.1589 + 0.1001 i
0.075 + 0.0646 i
Singularities of quadratic [2, 1, 1] approximant
2
-1.1589 - 0.1001 i
0.075 - 0.0646 i
3
2.2373
0.361
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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
-1.2627 + 0.2181 i
0.0441 + 0.0345 i
Singularities of quadratic [2, 2, 1] approximant
2
-1.2627 - 0.2181 i
0.0441 - 0.0345 i
3
1.9303
0.182
4
93.7711
35.4 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
-1.1049
0.0304
Singularities of quadratic [2, 2, 2] approximant
2
-2.1883
0.0444 i
3
2.1648 + 1.2207 i
0.178 + 0.0476 i
4
2.1648 - 1.2207 i
0.178 - 0.0476 i
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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
-1.0862
0.0261
Singularities of quadratic [3, 2, 2] approximant
2
1.797 + 1.1357 i
0.0992 + 0.00733 i
3
1.797 - 1.1357 i
0.0992 - 0.00733 i
4
-2.2522
0.043 i
5
10.8382
340.
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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
-0.986
0.0118
Singularities of quadratic [3, 3, 2] approximant
2
-1.2802 + 1.394 i
0.00207 + 0.0214 i
3
-1.2802 - 1.394 i
0.00207 - 0.0214 i
4
3.818
0.404
5
5.4312
0.246 i
6
-8.7222
0.121 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
-0.9874
0.0119
Singularities of quadratic [3, 3, 3] approximant
2
-1.2779 + 1.4235 i
0.00251 + 0.0215 i
3
-1.2779 - 1.4235 i
0.00251 - 0.0215 i
4
3.4787
0.81
5
8.1608
0.371 i
6
-16.4324
0.21 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
-0.4534 + 0.0001 i
0.000139 + 0.000139 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.4534 - 0.0001 i
0.000139 - 0.000139 i
3
-0.9682
0.00794
4
-1.5284 + 1.7687 i
0.0224 + 0.035 i
5
-1.5284 - 1.7687 i
0.0224 - 0.035 i
6
2.6717
3.77
7
12.3357
0.663 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
-0.5719 + 0.0006 i
0.00027 + 0.00027 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.5719 - 0.0006 i
0.00027 - 0.00027 i
3
-0.9597
0.00663
4
-1.6652 + 1.8943 i
0.0324 + 0.0419 i
5
-1.6652 - 1.8943 i
0.0324 - 0.0419 i
6
2.5812
1.61
7
11.038
0.694 i
8
-90.3944
620. 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
-0.9359
0.0056
Singularities of quadratic [4, 4, 4] approximant
2
-1.7122
0.0684 i
3
0.608 + 1.8105 i
0.0017 + 0.00492 i
4
0.608 - 1.8105 i
0.0017 - 0.00492 i
5
-0.563 + 2.018 i
0.00594 - 0.000859 i
6
-0.563 - 2.018 i
0.00594 + 0.000859 i
7
0.7721 + 2.4408 i
0.00777 + 0.000989 i
8
0.7721 - 2.4408 i
0.00777 - 0.000989 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
-0.9329
0.00545
Singularities of quadratic [5, 4, 4] approximant
2
-1.5313
0.0355 i
3
1.4531 + 1.1091 i
0.00429 + 0.0067 i
4
1.4531 - 1.1091 i
0.00429 - 0.0067 i
5
1.3198 + 1.3504 i
0.00732 - 0.00172 i
6
1.3198 - 1.3504 i
0.00732 + 0.00172 i
7
-0.8935 + 1.9145 i
0.0105 + 0.00307 i
8
-0.8935 - 1.9145 i
0.0105 - 0.00307 i
9
-2.9217
0.0198
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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
-0.9454
0.00901
Singularities of quadratic [5, 5, 4] approximant
2
-1.1146
0.0162 i
3
-1.2406
0.276
4
1.8321 + 1.1742 i
0.0204 + 0.0232 i
5
1.8321 - 1.1742 i
0.0204 - 0.0232 i
6
1.6248 + 1.7746 i
0.0297 - 0.00286 i
7
1.6248 - 1.7746 i
0.0297 + 0.00286 i
8
-1.5139 + 2.442 i
0.0306 + 0.023 i
9
-1.5139 - 2.442 i
0.0306 - 0.023 i
10
-8.223
0.768 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
-0.9097 + 0.0587 i
0.000421 + 0.00246 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.9097 - 0.0587 i
0.000421 - 0.00246 i
3
-0.9282
0.00181
4
2.1808 + 0.3425 i
0.228 + 0.0264 i
5
2.1808 - 0.3425 i
0.228 - 0.0264 i
6
3.0737
0.355
7
0.4939 + 3.3802 i
0.194 + 0.0116 i
8
0.4939 - 3.3802 i
0.194 - 0.0116 i
9
-3.1414 + 2.3606 i
0.0193 - 0.0869 i
10
-3.1414 - 2.3606 i
0.0193 + 0.0869 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
-0.9014 + 0.0521 i
0.0000811 + 0.00209 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.9014 - 0.0521 i
0.0000811 - 0.00209 i
3
-0.9056
0.00148
4
2.0985 + 0.7692 i
0.107 - 0.0329 i
5
2.0985 - 0.7692 i
0.107 + 0.0329 i
6
-3.0866
0.175 i
7
-0.98 + 3.1571 i
0.105 - 0.018 i
8
-0.98 - 3.1571 i
0.105 + 0.018 i
9
4.2845 + 1.7346 i
0.508 - 0.37 i
10
4.2845 - 1.7346 i
0.508 + 0.37 i
11
12.1233
0.827
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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
-0.9388 + 0.0539 i
0.00128 + 0.00512 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.9388 - 0.0539 i
0.00128 - 0.00512 i
3
-0.9899
0.00397
4
1.773
0.0175
5
1.8855
0.0232 i
6
2.1744 + 0.869 i
0.0586 - 0.0179 i
7
2.1744 - 0.869 i
0.0586 + 0.0179 i
8
1.9924 + 2.1041 i
0.0527 - 0.0252 i
9
1.9924 - 2.1041 i
0.0527 + 0.0252 i
10
-0.9407 + 2.9172 i
0.0542 - 0.00891 i
11
-0.9407 - 2.9172 i
0.0542 + 0.00891 i
12
-3.1895
0.239 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
0.0551
0
Singularities of quadratic [6, 6, 6] approximant
2
0.0551
0
3
-0.958 + 0.0436 i
0.00178 + 0.01 i
4
-0.958 - 0.0436 i
0.00178 - 0.01 i
5
-1.0345
0.007
6
2.1765 + 0.613 i
0.0327 - 0.191 i
7
2.1765 - 0.613 i
0.0327 + 0.191 i
8
-0.8058 + 3.5872 i
0.143 - 0.0579 i
9
-0.8058 - 3.5872 i
0.143 + 0.0579 i
10
-4.2269
0.146 i
11
4.2485 + 1.7011 i
0.113 + 0.648 i
12
4.2485 - 1.7011 i
0.113 - 0.648 i
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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
-0.9744 + 0.0256 i
0.0011 - 0.027 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.9744 - 0.0256 i
0.0011 + 0.027 i
3
-1.0769
0.0124
4
1.9711 + 0.7773 i
0.038 + 0.00239 i
5
1.9711 - 0.7773 i
0.038 - 0.00239 i
6
2.5659
0.117
7
-0.7433 + 2.9268 i
0.0369 - 0.00369 i
8
-0.7433 - 2.9268 i
0.0369 + 0.00369 i
9
1.5092 + 2.7264 i
0.0399 + 0.0126 i
10
1.5092 - 2.7264 i
0.0399 - 0.0126 i
11
-5.0463
0.521 i
12
-18.9312
4.78
13
34.3601
1.3 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
-0.9744 + 0.0253 i
0.00181 - 0.0272 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.9744 - 0.0253 i
0.00181 + 0.0272 i
3
-1.0766
0.0123
4
1.8652 + 0.6623 i
0.0133 - 0.00899 i
5
1.8652 - 0.6623 i
0.0133 + 0.00899 i
6
2.2284 + 0.1828 i
0.0224 - 0.00452 i
7
2.2284 - 0.1828 i
0.0224 + 0.00452 i
8
2.0133 + 2.2621 i
0.0348 - 0.0302 i
9
2.0133 - 2.2621 i
0.0348 + 0.0302 i
10
-0.6364 + 3.0066 i
0.0458 - 0.0202 i
11
-0.6364 - 3.0066 i
0.0458 + 0.0202 i
12
-3.6386 + 1.7731 i
0.0809 + 0.227 i
13
-3.6386 - 1.7731 i
0.0809 - 0.227 i
14
-13.751
0.443 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.