Singularities of Møller-Plesset series: example "o9"

Molecule N2. Basis cc-pVDZ. Structure ""

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Exampleso1o2o3o4o5o6o7o8o9
MoleculeNeNeF-HFH2OCH2CH2C2N2
Basiscc-pVDZcc-pVTZ-(f)cc-pVTZ-(f)cc-pVTZ-(f/d)cc-pVDZ(+)aug-cc-pVDZcc-pVTZ-(f/d)cc-pVDZ(+)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
-0.1659
0.0118
Singularities of quadratic [1, 1, 0] approximant
2
-0.202
0.013 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
-1.3811
0.691
Singularities of quadratic [1, 1, 1] approximant
2
2.8353
10.1
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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
-1.2094 + 0.9824 i
0.351 + 0.0607 i
Singularities of quadratic [2, 1, 1] approximant
2
-1.2094 - 0.9824 i
0.351 - 0.0607 i
3
1.6115
0.345
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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
1.9942
0.78
Singularities of quadratic [2, 2, 1] approximant
2
-1.86 + 0.8736 i
1.21 + 0.171 i
3
-1.86 - 0.8736 i
1.21 - 0.171 i
4
729.2936
24.6 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
1.9975
0.786
Singularities of quadratic [2, 2, 2] approximant
2
-1.8542 + 0.8731 i
1.2 + 0.18 i
3
-1.8542 - 0.8731 i
1.2 - 0.18 i
4
-748.1765
16.8
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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
0.2374
0.00232
Singularities of quadratic [3, 2, 2] approximant
2
0.2375
0.00232 i
3
-1.8356 + 0.9477 i
1.01 - 0.0998 i
4
-1.8356 - 0.9477 i
1.01 + 0.0998 i
5
2.0796
1.11
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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
-1.3369 + 0.9254 i
0.227 + 0.0195 i
Singularities of quadratic [3, 3, 2] approximant
2
-1.3369 - 0.9254 i
0.227 - 0.0195 i
3
1.762
0.339
4
-2.2078 + 1.4721 i
0.101 + 0.411 i
5
-2.2078 - 1.4721 i
0.101 - 0.411 i
6
12.0364
483. 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
1.8106
0.626
Singularities of quadratic [3, 3, 3] approximant
2
-1.6723 + 0.7292 i
0.959 + 0.839 i
3
-1.6723 - 0.7292 i
0.959 - 0.839 i
4
2.877
0.916 i
5
-4.077
1.15
6
5.9448
5.32
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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
1.7494
0.398
Singularities of quadratic [4, 3, 3] approximant
2
-1.6036 + 0.8087 i
0.706 + 0.194 i
3
-1.6036 - 0.8087 i
0.706 - 0.194 i
4
-2.8965
1.66
5
3.2263
0.92 i
6
-5.2801
2.48 i
7
6.1168
3.12
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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
-1.5417 + 0.0288 i
0.0889 + 0.074 i
Singularities of quadratic [4, 4, 3] approximant
2
-1.5417 - 0.0288 i
0.0889 - 0.074 i
3
-1.4942 + 0.739 i
0.213 + 0.192 i
4
-1.4942 - 0.739 i
0.213 - 0.192 i
5
1.7189
0.295
6
5.3194
3.14 i
7
-4.2112 + 7.6483 i
1.1 + 0.502 i
8
-4.2112 - 7.6483 i
1.1 - 0.502 i
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Table 13. 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
0.9546
0.008
Singularities of quadratic [4, 4, 4] approximant
2
0.9588
0.00799 i
3
1.6267
0.131
4
-1.6024 + 0.7419 i
0.433 + 0.641 i
5
-1.6024 - 0.7419 i
0.433 - 0.641 i
6
-3.7224 + 2.3496 i
0.0043 - 0.974 i
7
-3.7224 - 2.3496 i
0.0043 + 0.974 i
8
63.4829
0.991 i
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Table 14. 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
-1.5328 + 0.5331 i
0.243 - 0.103 i
Singularities of quadratic [5, 4, 4] approximant
2
-1.5328 - 0.5331 i
0.243 + 0.103 i
3
1.6579 + 0.3248 i
0.0931 - 0.1 i
4
1.6579 - 0.3248 i
0.0931 + 0.1 i
5
2.0224
0.164
6
-1.4867 + 1.3929 i
0.176 + 0.0986 i
7
-1.4867 - 1.3929 i
0.176 - 0.0986 i
8
-2.5504 + 1.2257 i
0.272 - 0.409 i
9
-2.5504 - 1.2257 i
0.272 + 0.409 i
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Exampleso1o2o3o4o5o6o7o8o9
MoleculeNeNeF-HFH2OCH2CH2C2N2
Basiscc-pVDZcc-pVTZ-(f)cc-pVTZ-(f)cc-pVTZ-(f/d)cc-pVDZ(+)aug-cc-pVDZcc-pVTZ-(f/d)cc-pVDZ(+)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.