Singularities of Møller-Plesset series: example "o8"

Molecule C2. 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 [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.3972
1.49
Singularities of quadratic [1, 0, 0] approximant
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Table 2. 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
-0.326
0.0592
Singularities of quadratic [1, 1, 0] approximant
2
-0.818
0.0938 i
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Table 3. 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
-0.7969
0.499
Singularities of quadratic [1, 1, 1] approximant
2
1.6932
2.98
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Table 4. 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
-0.9611
1.7
Singularities of quadratic [2, 1, 1] approximant
2
1.4562
0.85
3
-2.9074
0.832 i
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Table 5. 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
1.2435
0.446
Singularities of quadratic [2, 2, 1] approximant
2
-1.2758 + 0.3149 i
1.54 - 0.765 i
3
-1.2758 - 0.3149 i
1.54 + 0.765 i
4
129.4701
4.83 i
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Table 6. 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
-1.2018 + 0.3605 i
1.45 - 0.2 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.2018 - 0.3605 i
1.45 + 0.2 i
3
1.2731
0.493
4
-16.7027
312.
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Table 7. 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
-0.8005
0.141
Singularities of quadratic [3, 2, 2] approximant
2
-0.8487 + 0.2074 i
0.0791 + 0.201 i
3
-0.8487 - 0.2074 i
0.0791 - 0.201 i
4
1.2043
0.344
5
-1.432
3.14e3 i
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Table 8. 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
-0.8607 + 0.3816 i
0.205 + 0.0876 i
Singularities of quadratic [3, 3, 2] approximant
2
-0.8607 - 0.3816 i
0.205 - 0.0876 i
3
1.1833
0.304
4
-1.2658 + 0.3012 i
0.31 - 0.477 i
5
-1.2658 - 0.3012 i
0.31 + 0.477 i
6
309.2916
14.6 i
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Table 9. 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
0.0817
0.0000147
Singularities of quadratic [3, 3, 3] approximant
2
0.0817
0.0000147 i
3
-0.8143 + 0.1002 i
0.106 + 0.0428 i
4
-0.8143 - 0.1002 i
0.106 - 0.0428 i
5
1.1099
0.154
6
-3.1921
0.473
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Table 10. 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
-0.8959 + 0.1394 i
0.237 + 0.0481 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.8959 - 0.1394 i
0.237 - 0.0481 i
3
-0.1202 + 1.1617 i
0.0283 + 0.0689 i
4
-0.1202 - 1.1617 i
0.0283 - 0.0689 i
5
-0.282 + 1.2106 i
0.074 - 0.0217 i
6
-0.282 - 1.2106 i
0.074 + 0.0217 i
7
1.2765
0.746
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Table 11. 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
-0.9834 + 0.3334 i
0.528 + 0.765 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.9834 - 0.3334 i
0.528 - 0.765 i
3
1.2029 + 0.325 i
0.167 - 0.243 i
4
1.2029 - 0.325 i
0.167 + 0.243 i
5
1.8051
0.332
6
-1.9994
1.47
7
-3.2313
10.5 i
8
13.781
11.5 i
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Table 12. 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
-0.9378 + 0.2816 i
0.285 - 0.302 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.9378 - 0.2816 i
0.285 + 0.302 i
3
1.2257 + 0.3889 i
0.214 - 0.0909 i
4
1.2257 - 0.3889 i
0.214 + 0.0909 i
5
-1.2299 + 0.6983 i
0.266 - 0.0437 i
6
-1.2299 - 0.6983 i
0.266 + 0.0437 i
7
-1.661
0.816
8
2.638
1.81
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Table 13. 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
-0.9286 + 0.2674 i
0.287 - 0.182 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.9286 - 0.2674 i
0.287 + 0.182 i
3
1.2251 + 0.3912 i
0.213 - 0.0843 i
4
1.2251 - 0.3912 i
0.213 + 0.0843 i
5
-1.1815 + 0.6698 i
0.254 + 0.0104 i
6
-1.1815 - 0.6698 i
0.254 - 0.0104 i
7
-1.4412
2.58
8
2.6362
1.88
9
-73.4278
0.596 i
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Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.9744 + 0.3447 i
0.538 + 0.588 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.9744 - 0.3447 i
0.538 - 0.588 i
3
1.233 + 0.3392 i
0.208 - 0.258 i
4
1.233 - 0.3392 i
0.208 + 0.258 i
5
-1.8034
1.93
6
2.2204
0.497
7
-2.2484
65.7 i
8
5.5577
125. i
9
-6.858 + 2.9665 i
1.91 + 2.66 i
10
-6.858 - 2.9665 i
1.91 - 2.66 i
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Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.9743 + 0.3495 i
0.571 + 0.505 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.9743 - 0.3495 i
0.571 - 0.505 i
3
1.2301 + 0.346 i
0.208 - 0.228 i
4
1.2301 - 0.346 i
0.208 + 0.228 i
5
-1.5804
1.3
6
2.2875
0.6
7
-3.5535
0.48 i
8
-2.3429 + 3.0773 i
0.244 + 0.371 i
9
-2.3429 - 3.0773 i
0.244 - 0.371 i
10
91.1886
0.183 i
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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.9743 + 0.3469 i
0.546 + 0.548 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.9743 - 0.3469 i
0.546 - 0.548 i
3
1.2268 + 0.3521 i
0.209 - 0.2 i
4
1.2268 - 0.3521 i
0.209 + 0.2 i
5
-1.6616
1.4
6
2.2617
0.662
7
-3.307
0.976 i
8
-0.8513 + 3.4551 i
0.226 + 0.345 i
9
-0.8513 - 3.4551 i
0.226 - 0.345 i
10
0.7268 + 4.2062 i
0.414 - 0.297 i
11
0.7268 - 4.2062 i
0.414 + 0.297 i
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Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.8928
0.119
Singularities of quadratic [6, 6, 5] approximant
2
0.8935
0.117 i
3
-0.9751 + 0.348 i
0.58 + 0.538 i
4
-0.9751 - 0.348 i
0.58 - 0.538 i
5
1.2319 + 0.3374 i
0.214 - 0.266 i
6
1.2319 - 0.3374 i
0.214 + 0.266 i
7
-1.6619
1.47
8
2.1365
0.452
9
-2.4431
4.21 i
10
5.6796
78.4 i
11
-5.4737 + 3.1874 i
1.44 + 1.23 i
12
-5.4737 - 3.1874 i
1.44 - 1.23 i
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Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.5168
0.000729
Singularities of quadratic [6, 6, 6] approximant
2
0.5169
0.000729 i
3
-0.9792 + 0.3481 i
0.7 + 0.563 i
4
-0.9792 - 0.3481 i
0.7 - 0.563 i
5
1.2373 + 0.3574 i
0.295 - 0.153 i
6
1.2373 - 0.3574 i
0.295 + 0.153 i
7
-1.6058 + 0.2918 i
0.953 - 0.652 i
8
-1.6058 - 0.2918 i
0.953 + 0.652 i
9
1.8512
0.391
10
-2.2159
0.743
11
3.0572
1.08 i
12
11.9772
0.687
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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.0203 + 0.1757 i
0.0109 - 0.0131 i
Singularities of quadratic [7, 6, 6] approximant
2
1.0203 - 0.1757 i
0.0109 + 0.0131 i
3
-0.9845 + 0.3622 i
0.812 + 0.125 i
4
-0.9845 - 0.3622 i
0.812 - 0.125 i
5
1.0771 + 0.1805 i
0.0155 + 0.0126 i
6
1.0771 - 0.1805 i
0.0155 - 0.0126 i
7
-1.1789 + 0.0495 i
1.38 - 0.64 i
8
-1.1789 - 0.0495 i
1.38 + 0.64 i
9
1.191 + 0.4241 i
0.0707 + 0.0196 i
10
1.191 - 0.4241 i
0.0707 - 0.0196 i
11
-1.4141
0.797
12
4.3113
8.65
13
-18.5382
0.5 i
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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.5365 + 0.3368 i
0.000309 - 0.000699 i
Singularities of quadratic [7, 7, 6] approximant
2
0.5365 - 0.3368 i
0.000309 + 0.000699 i
3
0.5366 + 0.3368 i
0.000699 + 0.000309 i
4
0.5366 - 0.3368 i
0.000699 - 0.000309 i
5
-0.9854 + 0.35 i
0.949 + 0.521 i
6
-0.9854 - 0.35 i
0.949 - 0.521 i
7
1.2134 + 0.3419 i
0.105 - 0.183 i
8
1.2134 - 0.3419 i
0.105 + 0.183 i
9
-1.4532 + 0.2821 i
0.789 - 0.411 i
10
-1.4532 - 0.2821 i
0.789 + 0.411 i
11
-1.825
0.74
12
2.6175
1.23
13
-15.2158
1.19 i
14
16.2744
2.12 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.