Singularities of Møller-Plesset series: example "o3"

Molecule F-. Basis cc-pVTZ-(f). 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
-0.4351
0.0373
Singularities of quadratic [1, 1, 0] approximant
2
-0.6483
0.0456 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
-1.8087
0.671
Singularities of quadratic [1, 1, 1] approximant
2
4.9355
45.9
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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
-1.5549
0.322
Singularities of quadratic [2, 1, 1] approximant
2
2.5855 + 3.2621 i
0.743 + 0.225 i
3
2.5855 - 3.2621 i
0.743 - 0.225 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
-2.2237
5.05
Singularities of quadratic [2, 2, 1] approximant
2
2.8386
0.96
3
-4.0505
1.48 i
4
90.2541
4.32 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
-2.012
1.64
Singularities of quadratic [2, 2, 2] approximant
2
2.6096
0.765
3
7.26
1.28 i
4
-9.5994
0.781 i
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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
-1.9851
1.17
Singularities of quadratic [3, 2, 2] approximant
2
2.0338 + 0.1298 i
0.0498 - 0.0726 i
3
2.0338 - 0.1298 i
0.0498 + 0.0726 i
4
2.3362
0.103
5
-5.1074
1.84 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
-2.0947
3.28
Singularities of quadratic [3, 3, 2] approximant
2
2.4434
0.46
3
-4.4385 + 8.8953 i
0.503 - 0.368 i
4
-4.4385 - 8.8953 i
0.503 + 0.368 i
5
9.9435
2.13 i
6
-12.3749
0.547 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
-2.1747
13.9
Singularities of quadratic [3, 3, 3] approximant
2
2.2677
0.219
3
0.9268 + 3.5737 i
0.177 - 0.017 i
4
0.9268 - 3.5737 i
0.177 + 0.017 i
5
-0.0148 + 3.9484 i
0.0311 - 0.184 i
6
-0.0148 - 3.9484 i
0.0311 + 0.184 i
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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
-0.4332 + 0.e-4 i
0.000615 + 0.000615 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.4332 - 0.e-4 i
0.000615 - 0.000615 i
3
-2.0783
3.28
4
2.2047 + 0.5186 i
0.0805 - 0.118 i
5
2.2047 - 0.5186 i
0.0805 + 0.118 i
6
2.5886
0.131
7
-6.9014
1.16 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
0.4224
0.000224
Singularities of quadratic [4, 4, 3] approximant
2
0.4224
0.000224 i
3
-0.8618 + 0.0015 i
0.00374 + 0.00372 i
4
-0.8618 - 0.0015 i
0.00374 - 0.00372 i
5
-2.0847
5.48
6
2.2643
0.186
7
-5.8915
2.05 i
8
327.0342
20.8 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.