Singularities of Møller-Plesset series: example "a69"

Molecule HF. Basis cc-pVDZ. Structure "mpn_Rfci"

Content


Examplesa1a2a8a16a22a30a38a44a45a51a62a69a75a83a84a85a86a87a88a90a91
MoleculeArBHBHBHBHBHBHBO+C2CN+N2HFHFHClHClF-Cl-Cl-NeOH-SH-
Basisaug-cc-pVDZcc-pVDZcc-pVTZcc-pVQZaug-cc-pVDZaug-cc-pVTZaug-cc-pVQZcc-pVDZcc-pVDZcc-pVDZcc-pVDZcc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZaug-cc-pVDZaug-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
17.4809
7.05
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
0.5708
0.0399
Singularities of quadratic [1, 1, 0] approximant
2
0.8503
0.0487 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
-3.3094
1.41
Singularities of quadratic [1, 1, 1] approximant
2
3.6526
3.
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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
1.4969
0.0758
Singularities of quadratic [2, 1, 1] approximant
2
-1.2956 + 1.3496 i
0.0804 + 0.0117 i
3
-1.2956 - 1.3496 i
0.0804 - 0.0117 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
-1.7137
0.143
Singularities of quadratic [2, 2, 1] approximant
2
2.3823 + 1.699 i
0.341 + 0.0377 i
3
2.3823 - 1.699 i
0.341 - 0.0377 i
4
-9.7433
0.502 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
-1.8538 + 0.2686 i
0.249 + 0.195 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.8538 - 0.2686 i
0.249 - 0.195 i
3
2.4733
0.573
4
-5.7519
2.42
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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
-0.301
0.000195
Singularities of quadratic [3, 2, 2] approximant
2
-0.3012
0.000195 i
3
-1.3747
0.0301
4
2.2109
0.148
5
4.7852
11.5 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
-1.6621
0.181
Singularities of quadratic [3, 3, 2] approximant
2
2.3965
0.576
3
-3.5688
0.207 i
4
-0.0334 + 4.721 i
0.173 - 0.382 i
5
-0.0334 - 4.721 i
0.173 + 0.382 i
6
61.8124
299. 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
-1.537
0.0782
Singularities of quadratic [3, 3, 3] approximant
2
2.4644
0.719
3
-1.4533 + 2.1532 i
0.0489 + 0.0777 i
4
-1.4533 - 2.1532 i
0.0489 - 0.0777 i
5
-0.8128 + 2.6333 i
0.0777 - 0.0882 i
6
-0.8128 - 2.6333 i
0.0777 + 0.0882 i
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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
-1.5556
0.0908
Singularities of quadratic [4, 3, 3] approximant
2
-0.509 + 2.2766 i
0.0575 - 0.0525 i
3
-0.509 - 2.2766 i
0.0575 + 0.0525 i
4
2.3492
0.379
5
-1.0047 + 2.3365 i
0.0399 + 0.062 i
6
-1.0047 - 2.3365 i
0.0399 - 0.062 i
7
8.0894
11.5 i
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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
-1.3972
0.0332
Singularities of quadratic [4, 4, 3] approximant
2
-1.7312
0.046 i
3
2.6403
2.79
4
-2.5171 + 0.9666 i
0.0359 + 0.1 i
5
-2.5171 - 0.9666 i
0.0359 - 0.1 i
6
0.0669 + 3.5097 i
0.00164 + 0.159 i
7
0.0669 - 3.5097 i
0.00164 - 0.159 i
8
9013.5654
203. 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
-1.4458
0.0426
Singularities of quadratic [4, 4, 4] approximant
2
-2.2794
0.0978 i
3
-2.0268 + 1.3288 i
0.0298 + 0.0895 i
4
-2.0268 - 1.3288 i
0.0298 - 0.0895 i
5
2.7706
17.
6
0.4008 + 4.2354 i
0.0313 + 0.219 i
7
0.4008 - 4.2354 i
0.0313 - 0.219 i
8
6.0652
0.583 i
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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
1.4055
0.0343
Singularities of quadratic [5, 4, 4] approximant
2
1.4099
0.0347 i
3
-1.4267
0.0379
4
-1.9532
0.0664 i
5
2.549
1.2
6
-2.2967 + 1.2556 i
0.0361 + 0.0907 i
7
-2.2967 - 1.2556 i
0.0361 - 0.0907 i
8
-0.0521 + 3.6578 i
0.00341 - 0.174 i
9
-0.0521 - 3.6578 i
0.00341 + 0.174 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
-1.3722
0.0159
Singularities of quadratic [5, 5, 4] approximant
2
0.2797 + 1.4127 i
0.00125 + 0.000111 i
3
0.2797 - 1.4127 i
0.00125 - 0.000111 i
4
0.2882 + 1.4218 i
0.00011 - 0.00126 i
5
0.2882 - 1.4218 i
0.00011 + 0.00126 i
6
-2.1223
0.118 i
7
-1.4281 + 2.3975 i
0.0146 - 0.0339 i
8
-1.4281 - 2.3975 i
0.0146 + 0.0339 i
9
3.6719 + 1.0508 i
0.155 - 0.179 i
10
3.6719 - 1.0508 i
0.155 + 0.179 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
-1.1566 + 0.0031 i
0.000605 + 0.000612 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.1566 - 0.0031 i
0.000605 - 0.000612 i
3
-1.3291
0.00537
4
2.099 + 0.075 i
0.0372 - 0.0294 i
5
2.099 - 0.075 i
0.0372 + 0.0294 i
6
-2.6988
6.9 i
7
-1.5541 + 2.7724 i
0.0107 + 0.128 i
8
-1.5541 - 2.7724 i
0.0107 - 0.128 i
9
3.1826
0.65
10
122.3365
0.913 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
-1.3672
0.0174
Singularities of quadratic [6, 5, 5] approximant
2
-1.8326
0.0459 i
3
-1.5817 + 1.6119 i
0.015 + 0.0194 i
4
-1.5817 - 1.6119 i
0.015 - 0.0194 i
5
2.5655 + 0.6948 i
0.233 + 0.0835 i
6
2.5655 - 0.6948 i
0.233 - 0.0835 i
7
-2.4609 + 1.4587 i
0.044 - 0.0162 i
8
-2.4609 - 1.4587 i
0.044 + 0.0162 i
9
-0.2451 + 3.3329 i
0.012 - 0.0673 i
10
-0.2451 - 3.3329 i
0.012 + 0.0673 i
11
3.6728
3.07
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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
-1.24 + 0.0106 i
0.00114 + 0.00127 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.24 - 0.0106 i
0.00114 - 0.00127 i
3
-1.3245
0.00363
4
2.5005 + 0.3885 i
0.182 + 0.594 i
5
2.5005 - 0.3885 i
0.182 - 0.594 i
6
-2.6785
2.98 i
7
3.084
0.509
8
-1.8181 + 2.6489 i
0.0858 - 0.127 i
9
-1.8181 - 2.6489 i
0.0858 + 0.127 i
10
0.7762 + 5.7411 i
0.478 - 0.0863 i
11
0.7762 - 5.7411 i
0.478 + 0.0863 i
12
16.1916
1.3 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
-1.3013 + 0.0371 i
0.000427 + 0.0027 i
Singularities of quadratic [6, 6, 6] approximant
2
-1.3013 - 0.0371 i
0.000427 - 0.0027 i
3
-1.3115
0.00198
4
2.4785 + 0.311 i
0.522 + 0.323 i
5
2.4785 - 0.311 i
0.522 - 0.323 i
6
-2.7454
3.53 i
7
2.9607
0.543
8
-2.0807 + 2.6727 i
0.189 - 0.133 i
9
-2.0807 - 2.6727 i
0.189 + 0.133 i
10
-1.1453 + 5.2323 i
0.764 + 0.583 i
11
-1.1453 - 5.2323 i
0.764 - 0.583 i
12
-14.1423
3.98
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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
0.3706
6.57e-7
Singularities of quadratic [7, 6, 6] approximant
2
0.3706
6.57e-7 i
3
-1.2706
0.00103
4
-1.2843 + 0.0248 i
0.000308 - 0.00136 i
5
-1.2843 - 0.0248 i
0.000308 + 0.00136 i
6
2.5138 + 0.3717 i
0.0907 - 1.04 i
7
2.5138 - 0.3717 i
0.0907 + 1.04 i
8
2.786
0.497
9
-3.152
26.6 i
10
-2.027 + 3.2379 i
0.125 + 0.495 i
11
-2.027 - 3.2379 i
0.125 - 0.495 i
12
-3.4662 + 5.0077 i
5.24 + 3.17 i
13
-3.4662 - 5.0077 i
5.24 - 3.17 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
-1.2986 + 0.044 i
0.000144 + 0.00269 i
Singularities of quadratic [7, 7, 6] approximant
2
-1.2986 - 0.044 i
0.000144 - 0.00269 i
3
-1.2995
0.00191
4
2.4413 + 0.3113 i
0.246 + 0.0603 i
5
2.4413 - 0.3113 i
0.246 - 0.0603 i
6
3.0985 + 0.8629 i
0.291 + 0.0293 i
7
3.0985 - 0.8629 i
0.291 - 0.0293 i
8
-1.614 + 3.0946 i
0.0879 + 0.182 i
9
-1.614 - 3.0946 i
0.0879 - 0.182 i
10
-3.7137 + 0.7735 i
2.03 - 1.73 i
11
-3.7137 - 0.7735 i
2.03 + 1.73 i
12
4.9266
0.495
13
-5.7437
58.8 i
14
12.0981
3.25 i
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Examplesa1a2a8a16a22a30a38a44a45a51a62a69a75a83a84a85a86a87a88a90a91
MoleculeArBHBHBHBHBHBHBO+C2CN+N2HFHFHClHClF-Cl-Cl-NeOH-SH-
Basisaug-cc-pVDZcc-pVDZcc-pVTZcc-pVQZaug-cc-pVDZaug-cc-pVTZaug-cc-pVQZcc-pVDZcc-pVDZcc-pVDZcc-pVDZcc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZaug-cc-pVDZaug-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.