Singularities of Møller-Plesset series: example "o2"

Molecule Ne. 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
-0.1605
0.00647
Singularities of quadratic [1, 1, 0] approximant
2
-0.1785
0.00683 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
-1.9937
0.584
Singularities of quadratic [1, 1, 1] approximant
2
7.8597
51.2
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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
-1.0786
0.0434
Singularities of quadratic [2, 1, 1] approximant
2
0.7163 + 1.4331 i
0.0594 + 0.00679 i
3
0.7163 - 1.4331 i
0.0594 - 0.00679 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
-1.6704
0.31
Singularities of quadratic [2, 2, 1] approximant
2
5.3355 + 2.8566 i
1.13 + 1.17 i
3
5.3355 - 2.8566 i
1.13 - 1.17 i
4
-17.7119
1.07 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
0.5777 + 0.0009 i
0.00217 - 0.00217 i
Singularities of quadratic [2, 2, 2] approximant
2
0.5777 - 0.0009 i
0.00217 + 0.00217 i
3
-1.5528
0.168
4
3.3052
0.546
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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
-1.6335
0.334
Singularities of quadratic [3, 2, 2] approximant
2
3.2846
1.1
3
-2.0852 + 3.9416 i
0.237 - 0.494 i
4
-2.0852 - 3.9416 i
0.237 + 0.494 i
5
-5.2395
0.464 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
-1.6705
0.46
Singularities of quadratic [3, 3, 2] approximant
2
3.2425
0.926
3
-3.4775
0.385 i
4
-2.692 + 3.3672 i
0.488 - 0.409 i
5
-2.692 - 3.3672 i
0.488 + 0.409 i
6
5259.0761
87.8 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
-1.8616 + 0.1892 i
0.78 + 0.707 i
Singularities of quadratic [3, 3, 3] approximant
2
-1.8616 - 0.1892 i
0.78 - 0.707 i
3
3.2115
0.92
4
-3.2177
1.9
5
11.2609
2.68 i
6
-14.3388
1.14 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
0.3943
0.00131
Singularities of quadratic [4, 3, 3] approximant
2
0.3943
0.00131 i
3
-1.6763
0.494
4
3.3048
1.13
5
-3.5777
0.368 i
6
-2.4909 + 3.3804 i
0.397 - 0.386 i
7
-2.4909 - 3.3804 i
0.397 + 0.386 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
-1.6316
0.365
Singularities of quadratic [4, 4, 3] approximant
2
-0.1141 + 1.8709 i
0.00797 - 0.0317 i
3
-0.1141 - 1.8709 i
0.00797 + 0.0317 i
4
0.0093 + 1.9606 i
0.0324 + 0.006 i
5
0.0093 - 1.9606 i
0.0324 - 0.006 i
6
2.4883
0.119
7
6.4486
1.47 i
8
-34.2029
49. 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
-0.0212 + 1.1594 i
0.00345 - 0.00382 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.0212 - 1.1594 i
0.00345 + 0.00382 i
3
-0.0092 + 1.1655 i
0.00387 + 0.00342 i
4
-0.0092 - 1.1655 i
0.00387 - 0.00342 i
5
-1.7271
1.81
6
2.6743
0.155
7
-2.8558
0.351 i
8
-5.018
1.87
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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
0.0821
9.61e-8
Singularities of quadratic [5, 4, 4] approximant
2
0.0821
9.61e-8 i
3
-1.5803
0.169
4
2.0383
0.0271
5
-0.6984 + 2.2492 i
0.0591 + 0.0000469 i
6
-0.6984 - 2.2492 i
0.0591 - 0.0000469 i
7
-0.2307 + 2.8661 i
0.0286 + 0.0521 i
8
-0.2307 - 2.8661 i
0.0286 - 0.0521 i
9
2.946
0.066 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.