Singularities of Møller-Plesset series: example "a51"

Molecule CN+. Basis cc-pVDZ. Structure "mpn_Rfci"

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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

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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 [0, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [0, 0, 0] approximant
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Table 2. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [1, 0, 0] approximant
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Table 3. 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
-0.3529
0.0936
Singularities of quadratic [1, 1, 0] approximant
2
-1.3286
0.182 i
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Table 4. 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
-0.644
0.426
Singularities of quadratic [1, 1, 1] approximant
2
1.8943
6.15
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Table 5. 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
-0.5915
0.283
Singularities of quadratic [2, 1, 1] approximant
2
2.1713 + 1.1015 i
1.07 + 0.746 i
3
2.1713 - 1.1015 i
1.07 - 0.746 i
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Table 6. 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
-0.6155
0.363
Singularities of quadratic [2, 2, 1] approximant
2
1.5784
3.83
3
7.0566
0.767 i
4
-20.9446
1.12 i
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Table 7. 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
-0.6
0.288
Singularities of quadratic [2, 2, 2] approximant
2
0.7114 + 0.0215 i
0.0555 - 0.0539 i
3
0.7114 - 0.0215 i
0.0555 + 0.0539 i
4
1.3706
0.509
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Table 8. 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
-0.605
0.319
Singularities of quadratic [3, 2, 2] approximant
2
0.9674
7.36
3
0.9943
2.44 i
4
3.3588 + 0.2845 i
0.563 - 1.14 i
5
3.3588 - 0.2845 i
0.563 + 1.14 i
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Table 9. 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
-0.6258 + 0.1442 i
0.194 + 0.153 i
Singularities of quadratic [3, 3, 2] approximant
2
-0.6258 - 0.1442 i
0.194 - 0.153 i
3
-0.9332
0.494
4
1.1668
0.387
5
-2.186
1.81 i
6
343.1103
8.78 i
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Table 10. 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
-0.7245
2.75e5
Singularities of quadratic [3, 3, 3] approximant
2
-1.0528
0.261 i
3
-0.7961 + 0.7719 i
0.287 + 0.0731 i
4
-0.7961 - 0.7719 i
0.287 - 0.0731 i
5
1.1758
0.383
6
-2.095
0.369
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Table 11. 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
-0.6792 + 0.1303 i
0.413 + 0.22 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.6792 - 0.1303 i
0.413 - 0.22 i
3
1.0092
0.171
4
1.0308 + 0.1638 i
0.0738 + 0.201 i
5
1.0308 - 0.1638 i
0.0738 - 0.201 i
6
-1.4156 + 0.2959 i
2.2 + 2.29 i
7
-1.4156 - 0.2959 i
2.2 - 2.29 i
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Table 12. 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
-0.6765 + 0.1291 i
0.401 + 0.246 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.6765 - 0.1291 i
0.401 - 0.246 i
3
1.0699 + 0.2196 i
0.0383 + 0.264 i
4
1.0699 - 0.2196 i
0.0383 - 0.264 i
5
1.1387
0.192
6
-1.4391 + 0.1857 i
2.19 + 5.85 i
7
-1.4391 - 0.1857 i
2.19 - 5.85 i
8
2244.0148
32.2 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.