Singularities of Møller-Plesset series: example "a22"

Molecule BH. 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
0.8701
0.109
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
1.5638
0.423
Singularities of quadratic [1, 1, 0] approximant
2
13.4806
1.24 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
1.5073
0.347
Singularities of quadratic [1, 1, 1] approximant
2
10.1858
3.22 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
1.4338
0.255
Singularities of quadratic [2, 1, 1] approximant
2
5.6645
152. i
3
82.5
0.455
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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
1.4544
0.266
Singularities of quadratic [2, 2, 1] approximant
2
-2.1162 + 0.0536 i
0.0471 + 0.0451 i
3
-2.1162 - 0.0536 i
0.0471 - 0.0451 i
4
9.7057
2.4 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
1.8137 + 0.3749 i
0.445 + 0.161 i
Singularities of quadratic [2, 2, 2] approximant
2
1.8137 - 0.3749 i
0.445 - 0.161 i
3
5.5916
0.409
4
-9.5162
0.0563
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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
-1.0445
0.00181
Singularities of quadratic [3, 2, 2] approximant
2
-1.0564
0.00181 i
3
1.7833 + 0.5219 i
0.157 + 0.16 i
4
1.7833 - 0.5219 i
0.157 - 0.16 i
5
8.4471
0.18
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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
1.5631 + 0.4561 i
0.198 - 0.0407 i
Singularities of quadratic [3, 3, 2] approximant
2
1.5631 - 0.4561 i
0.198 + 0.0407 i
3
2.6465
1.78
4
-4.5406
0.0445
5
-6.3904
0.0508 i
6
27.407
0.332 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
1.5552 + 0.4108 i
0.209 - 0.143 i
Singularities of quadratic [3, 3, 3] approximant
2
1.5552 - 0.4108 i
0.209 + 0.143 i
3
2.4808
0.741
4
-4.409 + 0.5555 i
0.0224 + 0.0277 i
5
-4.409 - 0.5555 i
0.0224 - 0.0277 i
6
-28.6368
0.0357
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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
1.5555 + 0.4179 i
0.212 - 0.122 i
Singularities of quadratic [4, 3, 3] approximant
2
1.5555 - 0.4179 i
0.212 + 0.122 i
3
2.4697
0.76
4
-4.4127 + 0.3347 i
0.0445 + 0.0548 i
5
-4.4127 - 0.3347 i
0.0445 - 0.0548 i
6
33.8774
0.104 i
7
-37.7872
0.0755
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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
-1.1703 + 0.0006 i
0.000362 + 0.000362 i
Singularities of quadratic [4, 4, 3] approximant
2
-1.1703 - 0.0006 i
0.000362 - 0.000362 i
3
1.551 + 0.4007 i
0.175 - 0.174 i
4
1.551 - 0.4007 i
0.175 + 0.174 i
5
2.5953
0.976
6
-3.9367
0.0143
7
-6.8205
0.0229 i
8
54.4685
0.229 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
1.5435 + 0.374 i
0.0783 - 0.221 i
Singularities of quadratic [4, 4, 4] approximant
2
1.5435 - 0.374 i
0.0783 + 0.221 i
3
-3.1549 + 0.5398 i
0.000981 + 0.00231 i
4
-3.1549 - 0.5398 i
0.000981 - 0.00231 i
5
3.1595 + 0.9015 i
0.583 - 0.268 i
6
3.1595 - 0.9015 i
0.583 + 0.268 i
7
-3.8108
0.00212
8
8.3823
0.276
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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
1.5224 + 0.3586 i
0.00678 + 0.163 i
Singularities of quadratic [5, 4, 4] approximant
2
1.5224 - 0.3586 i
0.00678 - 0.163 i
3
-1.5746 + 0.0032 i
0.000168 + 0.000169 i
4
-1.5746 - 0.0032 i
0.000168 - 0.000169 i
5
2.4107 + 1.0948 i
0.0773 - 0.153 i
6
2.4107 - 1.0948 i
0.0773 + 0.153 i
7
-3.3033
0.00344
8
3.6328
1.37
9
-11.2095
0.0122 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
1.5881
0.222
Singularities of quadratic [5, 5, 4] approximant
2
1.5648 + 0.4511 i
0.149 + 0.00192 i
3
1.5648 - 0.4511 i
0.149 - 0.00192 i
4
1.7501
17.5 i
5
-3.2651
0.00799
6
3.3063
13.4
7
-4.2703
0.00746 i
8
-6.857 + 3.8139 i
0.0241 - 0.00313 i
9
-6.857 - 3.8139 i
0.0241 + 0.00313 i
10
12.8955
4.33 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
1.3746
0.0923
Singularities of quadratic [5, 5, 5] approximant
2
1.395
0.11 i
3
1.5462 + 0.4118 i
0.141 - 0.117 i
4
1.5462 - 0.4118 i
0.141 + 0.117 i
5
3.0318
11.8
6
-3.2193 + 0.3573 i
0.00185 + 0.00322 i
7
-3.2193 - 0.3573 i
0.00185 - 0.00322 i
8
-4.0827
0.00366
9
6.4938
0.588 i
10
110.78
0.0117
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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
1.6033 + 0.5038 i
0.061 + 0.0974 i
Singularities of quadratic [6, 5, 5] approximant
2
1.6033 - 0.5038 i
0.061 - 0.0974 i
3
1.6892
0.218
4
2.7404
0.324 i
5
-3.032
0.00245
6
3.0128 + 3.3024 i
0.048 + 0.00269 i
7
3.0128 - 3.3024 i
0.048 - 0.00269 i
8
-2.8597 + 3.622 i
0.000647 - 0.00269 i
9
-2.8597 - 3.622 i
0.000647 + 0.00269 i
10
-3.2444 + 5.2744 i
0.00355 + 0.00169 i
11
-3.2444 - 5.2744 i
0.00355 - 0.00169 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
1.6046
0.187
Singularities of quadratic [6, 6, 5] approximant
2
1.5661 + 0.4779 i
0.107 + 0.0412 i
3
1.5661 - 0.4779 i
0.107 - 0.0412 i
4
1.9195
2.95 i
5
-2.7304
0.00121
6
-3.1753
0.00104 i
7
-3.4951 + 1.1585 i
0.00252 - 0.000814 i
8
-3.4951 - 1.1585 i
0.00252 + 0.000814 i
9
4.3579
0.645
10
-6.9267
0.014
11
8.0308
3.27 i
12
-11.0333
9.61 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
1.5889
0.201
Singularities of quadratic [6, 6, 6] approximant
2
1.5591 + 0.4646 i
0.121 + 0.0123 i
3
1.5591 - 0.4646 i
0.121 - 0.0123 i
4
1.8089
141. i
5
-2.5621 + 0.0434 i
0.000349 + 0.000397 i
6
-2.5621 - 0.0434 i
0.000349 - 0.000397 i
7
-3.0001
0.00105
8
3.7136
1.6
9
1.8588 + 4.7733 i
0.0144 - 0.0111 i
10
1.8588 - 4.7733 i
0.0144 + 0.0111 i
11
0.7549 + 5.1224 i
0.00645 + 0.0096 i
12
0.7549 - 5.1224 i
0.00645 - 0.0096 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
1.5891
0.201
Singularities of quadratic [7, 6, 6] approximant
2
1.559 + 0.4646 i
0.121 + 0.012 i
3
1.559 - 0.4646 i
0.121 - 0.012 i
4
1.809
133. i
5
-2.5497 + 0.0415 i
0.000339 + 0.000382 i
6
-2.5497 - 0.0415 i
0.000339 - 0.000382 i
7
-2.9967
0.00105
8
3.724
1.56
9
1.8547 + 4.7067 i
0.0136 - 0.0116 i
10
1.8547 - 4.7067 i
0.0136 + 0.0116 i
11
0.7965 + 5.0787 i
0.00674 + 0.00931 i
12
0.7965 - 5.0787 i
0.00674 - 0.00931 i
13
589.5998
0.0232 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
1.6019
0.217
Singularities of quadratic [7, 7, 6] approximant
2
1.5561 + 0.4684 i
0.109 + 0.00956 i
3
1.5561 - 0.4684 i
0.109 - 0.00956 i
4
1.8581
6.2 i
5
-2.0094 + 0.0049 i
0.0000832 + 0.0000837 i
6
-2.0094 - 0.0049 i
0.0000832 - 0.0000837 i
7
-2.9293
0.0011
8
0.2655 + 3.0244 i
0.00212 - 0.000269 i
9
0.2655 - 3.0244 i
0.00212 + 0.000269 i
10
0.3517 + 3.04 i
0.00029 + 0.00224 i
11
0.3517 - 3.04 i
0.00029 - 0.00224 i
12
4.414
0.703
13
14.0665
0.214 i
14
-76.1998
0.0656 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.