Singularities of Møller-Plesset series: example "o4"

Molecule HF. Basis cc-pVTZ-(f/d). 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
-0.1506
0.0068
Singularities of quadratic [1, 1, 0] approximant
2
-0.1704
0.00723 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
-1.7346
0.584
Singularities of quadratic [1, 1, 1] approximant
2
5.0764
527.
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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
-1.0433
0.0643
Singularities of quadratic [2, 1, 1] approximant
2
0.7594 + 1.4229 i
0.0906 + 0.00984 i
3
0.7594 - 1.4229 i
0.0906 - 0.00984 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
-1.5561
0.417
Singularities of quadratic [2, 2, 1] approximant
2
3.8669
53.1
3
12.3549
1.31 i
4
-14.9908
0.965 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
0.5717 + 0.0021 i
0.0054 - 0.00538 i
Singularities of quadratic [2, 2, 2] approximant
2
0.5717 - 0.0021 i
0.0054 + 0.00538 i
3
-1.4466
0.23
4
2.9308
0.801
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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
-1.416 + 0.1652 i
0.188 + 0.162 i
Singularities of quadratic [3, 2, 2] approximant
2
-1.416 - 0.1652 i
0.188 - 0.162 i
3
-2.3349
0.973
4
2.5789
0.681
5
-3.7513
8.4 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
-1.2161 + 0.1558 i
0.0236 + 0.0399 i
Singularities of quadratic [3, 3, 2] approximant
2
-1.2161 - 0.1558 i
0.0236 - 0.0399 i
3
-1.3931
0.0453
4
2.4802
0.5
5
-5.1513
4.96 i
6
297.2105
19.6 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
-1.2458
0.121
Singularities of quadratic [3, 3, 3] approximant
2
-1.6405
0.182 i
3
-2.401
11.3
4
2.4221 + 0.8107 i
0.256 - 0.126 i
5
2.4221 - 0.8107 i
0.256 + 0.126 i
6
4.2288
0.88
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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
-0.9824 + 0.0175 i
0.00654 + 0.00695 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.9824 - 0.0175 i
0.00654 - 0.00695 i
3
-1.1939
0.0265
4
2.493
0.504
5
-3.423
1.9 i
6
-5.4278 + 1.6789 i
1. + 0.972 i
7
-5.4278 - 1.6789 i
1. - 0.972 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
-0.0519
4.75e-8 + 4.75e-8 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.0519
4.75e-8 - 4.75e-8 i
3
-0.6221 + 0.001 i
0.000435 + 0.000434 i
4
-0.6221 - 0.001 i
0.000435 - 0.000434 i
5
-1.1611
0.0262
6
2.5476
0.712
7
-4.2108
144. i
8
175.0269
21.6 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
-1.2735
0.325
Singularities of quadratic [4, 4, 4] approximant
2
-1.447
0.279 i
3
0.3243 + 1.7495 i
0.0394 + 0.00286 i
4
0.3243 - 1.7495 i
0.0394 - 0.00286 i
5
0.2342 + 1.7702 i
0.00468 - 0.0397 i
6
0.2342 - 1.7702 i
0.00468 + 0.0397 i
7
-2.1318
30.5
8
2.2735
0.213
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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
-0.0472
1.81e-9 + 1.81e-9 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.0472
1.81e-9 - 1.81e-9 i
3
0.0821 + 0.9505 i
0.000759 - 0.00132 i
4
0.0821 - 0.9505 i
0.000759 + 0.00132 i
5
0.0756 + 0.9548 i
0.00132 + 0.000786 i
6
0.0756 - 0.9548 i
0.00132 - 0.000786 i
7
-1.1498
0.0259
8
2.3405
0.216
9
-4.4604
7.42 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.