Singularities of Møller-Plesset series: example "a38"

Molecule BH. Basis aug-cc-pVQZ. 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
1.3265
0.209
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
1.2654
0.19
Singularities of quadratic [1, 1, 0] approximant
2
2330.1933
8.15 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
1.5034
0.366
Singularities of quadratic [1, 1, 1] approximant
2
-8.019
0.922
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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
1.4651
0.327
Singularities of quadratic [2, 1, 1] approximant
2
-4.851
0.591
3
-10.0544
0.378 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
1.4514
0.304
Singularities of quadratic [2, 2, 1] approximant
2
-5.9008
1.21
3
-12.6702
0.548 i
4
409.7523
2.76 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
1.4865
0.397
Singularities of quadratic [2, 2, 2] approximant
2
4.1292
0.718 i
3
-5.6066
0.454
4
5.8887
2.52
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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
0.8396
0.0291
Singularities of quadratic [3, 2, 2] approximant
2
0.8483
0.0289 i
3
1.3991
0.192
4
-5.1956
0.442
5
-18.713
0.365 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
1.5944 + 0.3268 i
0.312 - 0.153 i
Singularities of quadratic [3, 3, 2] approximant
2
1.5944 - 0.3268 i
0.312 + 0.153 i
3
2.3197
0.851
4
-3.8451
0.119
5
-12.803
0.179 i
6
18.0688
0.357 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
1.5886 + 0.3256 i
0.304 - 0.16 i
Singularities of quadratic [3, 3, 3] approximant
2
1.5886 - 0.3256 i
0.304 + 0.16 i
3
2.2805
0.759
4
-3.8327
0.119
5
-11.7935
0.176 i
6
21.4977
0.352 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
1.6162 + 0.3654 i
0.274 - 0.0141 i
Singularities of quadratic [4, 3, 3] approximant
2
1.6162 - 0.3654 i
0.274 + 0.0141 i
3
2.6238
5.7
4
-3.5012
0.062
5
-1.5659 + 8.783 i
0.0332 + 0.119 i
6
-1.5659 - 8.783 i
0.0332 - 0.119 i
7
-76.3262
0.878 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
1.6056 + 0.3443 i
0.306 - 0.0756 i
Singularities of quadratic [4, 4, 3] approximant
2
1.6056 - 0.3443 i
0.306 + 0.0756 i
3
-1.8411
0.011
4
-1.8583
0.0111 i
5
2.4182
1.38
6
-4.2886
0.357
7
-10.0149
0.178 i
8
22.3904
0.368 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
1.5973 + 0.3038 i
0.302 - 0.285 i
Singularities of quadratic [4, 4, 4] approximant
2
1.5973 - 0.3038 i
0.302 + 0.285 i
3
-2.0892 + 0.0178 i
0.00762 + 0.00774 i
4
-2.0892 - 0.0178 i
0.00762 - 0.00774 i
5
2.4079
0.799
6
-3.694
0.0586
7
5.9493
0.767 i
8
31.8549
1.46
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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
1.5942 + 0.2933 i
0.255 - 0.347 i
Singularities of quadratic [5, 4, 4] approximant
2
1.5942 - 0.2933 i
0.255 + 0.347 i
3
-1.762 + 0.0075 i
0.00312 + 0.00313 i
4
-1.762 - 0.0075 i
0.00312 - 0.00313 i
5
2.4553
0.8
6
-3.5267
0.045
7
5.3396
0.851 i
8
11.14 + 15.0748 i
0.408 + 0.405 i
9
11.14 - 15.0748 i
0.408 - 0.405 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
1.3587 + 0.0095 i
0.081 - 0.0901 i
Singularities of quadratic [5, 5, 4] approximant
2
1.3587 - 0.0095 i
0.081 + 0.0901 i
3
1.6448 + 0.3601 i
0.345 + 0.0715 i
4
1.6448 - 0.3601 i
0.345 - 0.0715 i
5
-2.3256
0.0183
6
-2.4077
0.0187 i
7
2.5195
2.92
8
-4.9147
5.48
9
-8.3908
0.192 i
10
26.575
0.393 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
1.0025 + 0.0001 i
0.0295 - 0.0297 i
Singularities of quadratic [5, 5, 5] approximant
2
1.0025 - 0.0001 i
0.0295 + 0.0297 i
3
1.6074 + 0.3297 i
0.334 - 0.135 i
4
1.6074 - 0.3297 i
0.334 + 0.135 i
5
-2.1751 + 0.0114 i
0.0271 + 0.0282 i
6
-2.1751 - 0.0114 i
0.0271 - 0.0282 i
7
2.4342
1.17
8
-3.9298
0.103
9
9.0674
0.425 i
10
-44.7208
0.208 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
1.4724 + 0.227 i
0.0412 + 0.0306 i
Singularities of quadratic [6, 5, 5] approximant
2
1.4724 - 0.227 i
0.0412 - 0.0306 i
3
1.399 + 0.7826 i
0.011 - 0.00462 i
4
1.399 - 0.7826 i
0.011 + 0.00462 i
5
1.4898 + 0.7394 i
0.00664 + 0.0123 i
6
1.4898 - 0.7394 i
0.00664 - 0.0123 i
7
-2.0214 + 0.0344 i
0.00182 + 0.0018 i
8
-2.0214 - 0.0344 i
0.00182 - 0.0018 i
9
-3.3883
0.0203
10
4.4825
0.416
11
-12.1587
0.763 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
1.4792 + 0.246 i
0.0351 + 0.0418 i
Singularities of quadratic [6, 6, 5] approximant
2
1.4792 - 0.246 i
0.0351 - 0.0418 i
3
1.3152 + 0.7784 i
0.00715 - 0.00355 i
4
1.3152 - 0.7784 i
0.00715 + 0.00355 i
5
1.3682 + 0.7509 i
0.00435 + 0.00746 i
6
1.3682 - 0.7509 i
0.00435 - 0.00746 i
7
-2.0343 + 0.0358 i
0.00189 + 0.00187 i
8
-2.0343 - 0.0358 i
0.00189 - 0.00187 i
9
-3.4166
0.0214
10
4.0852
0.553
11
-12.0568
0.653 i
12
2842.6613
8.3 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
0.3117
1.25e-7
Singularities of quadratic [6, 6, 6] approximant
2
0.3117
1.25e-7 i
3
1.3571 + 0.2802 i
0.0043 + 0.00579 i
4
1.3571 - 0.2802 i
0.0043 - 0.00579 i
5
1.347 + 0.6345 i
0.00201 + 0.00264 i
6
1.347 - 0.6345 i
0.00201 - 0.00264 i
7
1.2963 + 0.7464 i
0.00312 - 0.000908 i
8
1.2963 - 0.7464 i
0.00312 + 0.000908 i
9
-2.046
0.00278
10
-2.1349
0.00306 i
11
-3.9782
0.282
12
6.8139
0.421
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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
1.3403
0.00869
Singularities of quadratic [7, 6, 6] approximant
2
0.7509 + 1.26 i
0.000301 + 0.000151 i
3
0.7509 - 1.26 i
0.000301 - 0.000151 i
4
0.7507 + 1.282 i
0.000158 - 0.000303 i
5
0.7507 - 1.282 i
0.000158 + 0.000303 i
6
1.4416 + 0.5718 i
0.00304 + 0.00613 i
7
1.4416 - 0.5718 i
0.00304 - 0.00613 i
8
1.899
0.457 i
9
-2.0796
0.00125
10
-2.313
0.00154 i
11
1.6087 + 3.981 i
0.00283 - 0.0166 i
12
1.6087 - 3.981 i
0.00283 + 0.0166 i
13
-7.842
0.0545
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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
-0.4363 + 0.e-4 i
3.44e-8 + 3.44e-8 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.4363 - 0.e-4 i
3.44e-8 - 3.44e-8 i
3
0.9417 + 0.0008 i
0.000044 - 0.0000438 i
4
0.9417 - 0.0008 i
0.000044 + 0.0000438 i
5
1.4007
0.0119
6
-0.5477 + 1.7257 i
0.000157 - 0.0000204 i
7
-0.5477 - 1.7257 i
0.000157 + 0.0000204 i
8
-0.5898 + 1.7378 i
0.0000108 + 0.00016 i
9
-0.5898 - 1.7378 i
0.0000108 - 0.00016 i
10
1.753 + 0.7754 i
0.0184 + 0.0153 i
11
1.753 - 0.7754 i
0.0184 - 0.0153 i
12
-2.0694
0.000459
13
-2.8194
0.00141 i
14
86.723
0.76 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

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