Singularities of Møller-Plesset series: example "a8"

Molecule BH. Basis cc-pVTZ. 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 [3, 3, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.5733 + 0.3243 i
0.269 - 0.237 i
Singularities of quadratic [3, 3, 2] approximant
2
1.5733 - 0.3243 i
0.269 + 0.237 i
3
2.1857
0.493
4
-6.0635
0.181
5
-23.5914
0.215 i
6
1245.2983
1.93 i
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Table 2. 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
1.5729 + 0.3239 i
0.268 - 0.239 i
Singularities of quadratic [3, 3, 3] approximant
2
1.5729 - 0.3239 i
0.268 + 0.239 i
3
2.1827
0.489
4
-6.0709
0.184
5
-21.9614
0.224 i
6
-418.9354
2.16
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Table 3. 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
0.5746 + 0.e-4 i
0.000897 - 0.000897 i
Singularities of quadratic [4, 3, 3] approximant
2
0.5746 - 0.e-4 i
0.000897 + 0.000897 i
3
1.5505 + 0.2966 i
0.128 - 0.332 i
4
1.5505 - 0.2966 i
0.128 + 0.332 i
5
2.1218
0.39
6
-5.8007
0.14
7
-28.5096
0.192 i
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Table 4. 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
1.5902 + 0.366 i
0.272 - 0.0715 i
Singularities of quadratic [4, 4, 3] approximant
2
1.5902 - 0.366 i
0.272 + 0.0715 i
3
2.346
0.97
4
-3.8027
0.026
5
-4.9206
0.0275 i
6
-4.6869 + 7.0614 i
0.0772 - 0.0877 i
7
-4.6869 - 7.0614 i
0.0772 + 0.0877 i
8
33.7868
5.52 i
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Table 5. 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
1.5912 + 0.3735 i
0.27 - 0.029 i
Singularities of quadratic [4, 4, 4] approximant
2
1.5912 - 0.3735 i
0.27 + 0.029 i
3
2.248
0.772
4
-0.6592 + 2.4583 i
0.00443 + 0.0109 i
5
-0.6592 - 2.4583 i
0.00443 - 0.0109 i
6
-0.679 + 2.5177 i
0.0113 - 0.00419 i
7
-0.679 - 2.5177 i
0.0113 + 0.00419 i
8
-4.9634
0.0503
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Table 6. 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
1.5787 + 0.2702 i
0.0797 + 0.372 i
Singularities of quadratic [5, 4, 4] approximant
2
1.5787 - 0.2702 i
0.0797 - 0.372 i
3
2.2061 + 0.727 i
0.224 + 0.211 i
4
2.2061 - 0.727 i
0.224 - 0.211 i
5
-2.4712 + 0.0271 i
0.00306 + 0.00308 i
6
-2.4712 - 0.0271 i
0.00306 - 0.00308 i
7
3.2013
0.561
8
-4.9032
0.0325
9
16.4278
0.564 i
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Table 7. 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
1.3998 + 0.0329 i
0.0394 - 0.0367 i
Singularities of quadratic [5, 5, 4] approximant
2
1.3998 - 0.0329 i
0.0394 + 0.0367 i
3
1.6915 + 0.4481 i
0.0679 + 0.341 i
4
1.6915 - 0.4481 i
0.0679 - 0.341 i
5
2.1807
1.16
6
-3.7952 + 0.0761 i
0.102 + 0.0968 i
7
-3.7952 - 0.0761 i
0.102 - 0.0968 i
8
-9.4334
1.08
9
-17.5377
1.01 i
10
34.1665
1.23 i
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Table 8. 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
1.4097 + 0.0365 i
0.0416 - 0.038 i
Singularities of quadratic [5, 5, 5] approximant
2
1.4097 - 0.0365 i
0.0416 + 0.038 i
3
1.6983 + 0.4525 i
0.0356 + 0.345 i
4
1.6983 - 0.4525 i
0.0356 - 0.345 i
5
2.1646
1.15
6
-3.8295 + 0.0977 i
0.0787 + 0.0736 i
7
-3.8295 - 0.0977 i
0.0787 - 0.0736 i
8
-10.7667 + 2.519 i
0.434 + 1.18 i
9
-10.7667 - 2.519 i
0.434 - 1.18 i
10
66.1385
0.427 i
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Table 9. 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
1.4258 + 0.0499 i
0.038 - 0.0309 i
Singularities of quadratic [6, 5, 5] approximant
2
1.4258 - 0.0499 i
0.038 + 0.0309 i
3
1.7378 + 0.4815 i
0.17 - 0.267 i
4
1.7378 - 0.4815 i
0.17 + 0.267 i
5
1.9955
0.743
6
-3.3634 + 0.2346 i
0.00327 + 0.00418 i
7
-3.3634 - 0.2346 i
0.00327 - 0.00418 i
8
-4.2643
0.00754
9
-1.0869 + 10.6297 i
0.144 + 0.088 i
10
-1.0869 - 10.6297 i
0.144 - 0.088 i
11
-13.2713
0.121 i
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Table 10. 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
1.4444 + 0.0692 i
0.039 - 0.0286 i
Singularities of quadratic [6, 6, 5] approximant
2
1.4444 - 0.0692 i
0.039 + 0.0286 i
3
1.7233 + 0.5483 i
0.108 - 0.124 i
4
1.7233 - 0.5483 i
0.108 + 0.124 i
5
1.8216
0.528
6
2.862
0.678 i
7
-3.6866 + 0.2224 i
0.015 + 0.0146 i
8
-3.6866 - 0.2224 i
0.015 - 0.0146 i
9
3.7212
1.25
10
-6.8583
0.086
11
18.0958
2.98 i
12
-27.5978
1.28 i
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Table 11. 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
0.3727 + 0.e-5 i
1.44e-7 - 1.44e-7 i
Singularities of quadratic [6, 6, 6] approximant
2
0.3727 - 0.e-5 i
1.44e-7 + 1.44e-7 i
3
1.1773 + 0.0065 i
0.000619 - 0.0006 i
4
1.1773 - 0.0065 i
0.000619 + 0.0006 i
5
1.4178
0.0127
6
1.6929 + 0.6798 i
0.0406 - 0.00596 i
7
1.6929 - 0.6798 i
0.0406 + 0.00596 i
8
2.8668
0.496 i
9
-3.44
0.00855
10
-4.4027
0.0123 i
11
4.9751
1.09
12
-14.3294
0.222
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Table 12. 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
1.3452 + 0.1314 i
0.0405 + 0.0124 i
Singularities of quadratic [7, 6, 6] approximant
2
1.3452 - 0.1314 i
0.0405 - 0.0124 i
3
1.3586 + 0.107 i
0.0153 - 0.0388 i
4
1.3586 - 0.107 i
0.0153 + 0.0388 i
5
1.6981 + 0.4596 i
0.0258 - 0.341 i
6
1.6981 - 0.4596 i
0.0258 + 0.341 i
7
2.0846
0.786
8
-3.609 + 0.2265 i
0.0104 + 0.0114 i
9
-3.609 - 0.2265 i
0.0104 - 0.0114 i
10
-5.6194
0.0335
11
-0.0007 + 13.8331 i
0.235 + 0.203 i
12
-0.0007 - 13.8331 i
0.235 - 0.203 i
13
-35.1917
1.33 i
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Table 13. 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
1.4624
0.0386
Singularities of quadratic [7, 7, 6] approximant
2
1.4272 + 0.3745 i
0.0121 + 0.00503 i
3
1.4272 - 0.3745 i
0.0121 - 0.00503 i
4
1.4101 + 0.439 i
0.00685 - 0.0103 i
5
1.4101 - 0.439 i
0.00685 + 0.0103 i
6
1.9162 + 0.6441 i
0.0472 + 0.0992 i
7
1.9162 - 0.6441 i
0.0472 - 0.0992 i
8
-3.3868
0.00702
9
-4.5445
0.00859 i
10
-5.7371 + 4.805 i
0.00142 - 0.0249 i
11
-5.7371 - 4.805 i
0.00142 + 0.0249 i
12
-4.0523 + 7.3331 i
0.0313 - 0.0184 i
13
-4.0523 - 7.3331 i
0.0313 + 0.0184 i
14
14.9408
13.8 i
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Table 14. Singularities with their weights for the quadratic approximant [7, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.4135 + 0.3242 i
0.016 - 0.00102 i
Singularities of quadratic [7, 7, 7] approximant
2
1.4135 - 0.3242 i
0.016 + 0.00102 i
3
1.4775
0.0412
4
1.4279 + 0.3922 i
0.00271 - 0.0172 i
5
1.4279 - 0.3922 i
0.00271 + 0.0172 i
6
1.9147 + 0.5667 i
0.097 + 0.135 i
7
1.9147 - 0.5667 i
0.097 - 0.135 i
8
-3.4238
0.00911
9
-4.305
0.0106 i
10
-2.9225 + 7.5001 i
0.0319 - 0.034 i
11
-2.9225 - 7.5001 i
0.0319 + 0.034 i
12
-6.9508 + 6.2503 i
0.00734 + 0.0431 i
13
-6.9508 - 6.2503 i
0.00734 - 0.0431 i
14
37.4998
0.711 i
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Table 15. Singularities with their weights for the quadratic approximant [8, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.1954 + 0.0083 i
0.00137 - 0.00146 i
Singularities of quadratic [8, 7, 7] approximant
2
1.1954 - 0.0083 i
0.00137 + 0.00146 i
3
1.2766 + 0.1515 i
0.00469 + 0.00171 i
4
1.2766 - 0.1515 i
0.00469 - 0.00171 i
5
1.2811 + 0.1856 i
0.00272 - 0.00609 i
6
1.2811 - 0.1856 i
0.00272 + 0.00609 i
7
1.76 + 0.4299 i
0.627 - 0.315 i
8
1.76 - 0.4299 i
0.627 + 0.315 i
9
1.8633
0.314
10
-3.6361 + 0.1893 i
0.0147 + 0.016 i
11
-3.6361 - 0.1893 i
0.0147 - 0.016 i
12
-5.927
0.0497
13
0.5443 + 13.592 i
0.21 + 0.246 i
14
0.5443 - 13.592 i
0.21 - 0.246 i
15
-124.2932
6.42 i
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Table 16. Singularities with their weights for the quadratic approximant [8, 8, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.6039 + 0.e-5 i
1.23e-6 + 1.23e-6 i
Singularities of quadratic [8, 8, 7] approximant
2
-0.6039 - 0.e-5 i
1.23e-6 - 1.23e-6 i
3
1.4172 + 0.348 i
0.014 + 0.00213 i
4
1.4172 - 0.348 i
0.014 - 0.00213 i
5
1.4683
0.0391
6
1.4164 + 0.4145 i
0.00508 - 0.0133 i
7
1.4164 - 0.4145 i
0.00508 + 0.0133 i
8
1.9151 + 0.6057 i
0.0702 + 0.116 i
9
1.9151 - 0.6057 i
0.0702 - 0.116 i
10
-3.419
0.00897
11
-4.3162
0.0101 i
12
-6.9693 + 4.4635 i
0.00871 - 0.0417 i
13
-6.9693 - 4.4635 i
0.00871 + 0.0417 i
14
-4.2617 + 8.1396 i
0.0458 - 0.0331 i
15
-4.2617 - 8.1396 i
0.0458 + 0.0331 i
16
16.9624
64.4 i
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Table 17. Singularities with their weights for the quadratic approximant [8, 8, 8]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.4354 + 0.2262 i
0.0103 + 0.0184 i
Singularities of quadratic [8, 8, 8] approximant
2
1.4354 - 0.2262 i
0.0103 - 0.0184 i
3
1.4474 + 0.3615 i
0.0148 + 0.00587 i
4
1.4474 - 0.3615 i
0.0148 - 0.00587 i
5
1.8051
0.382
6
1.2801 + 1.4346 i
0.00053 + 0.00109 i
7
1.2801 - 1.4346 i
0.00053 - 0.00109 i
8
1.2757 + 1.4492 i
0.00111 - 0.000508 i
9
1.2757 - 1.4492 i
0.00111 + 0.000508 i
10
1.8762 + 0.6776 i
0.00974 + 0.0409 i
11
1.8762 - 0.6776 i
0.00974 - 0.0409 i
12
-2.0078
0.0000684
13
-2.0106
0.0000684 i
14
-3.1907
0.00189
15
-6.0938
0.0144 i
16
53.7328
0.159 i
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Table 18. Singularities with their weights for the quadratic approximant [9, 8, 8]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.1726 + 1.2719 i
6.23e-6 - 0.0000297 i
Singularities of quadratic [9, 8, 8] approximant
2
-0.1726 - 1.2719 i
6.23e-6 + 0.0000297 i
3
-0.1726 + 1.2719 i
0.0000297 + 6.23e-6 i
4
-0.1726 - 1.2719 i
0.0000297 - 6.23e-6 i
5
1.4218 + 0.3148 i
0.0187 - 0.00382 i
6
1.4218 - 0.3148 i
0.0187 + 0.00382 i
7
1.4369 + 0.388 i
0.000648 - 0.0199 i
8
1.4369 - 0.388 i
0.000648 + 0.0199 i
9
1.4894
0.0499
10
1.9222 + 0.5557 i
0.0918 + 0.142 i
11
1.9222 - 0.5557 i
0.0918 - 0.142 i
12
-3.2629 + 0.4468 i
0.00131 + 0.00167 i
13
-3.2629 - 0.4468 i
0.00131 - 0.00167 i
14
-3.9332
0.00251
15
-13.9029
0.251 i
16
2.3546 + 19.5756 i
0.521 + 0.0917 i
17
2.3546 - 19.5756 i
0.521 - 0.0917 i
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Table 19. Singularities with their weights for the quadratic approximant [9, 9, 8]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.7579 + 0.6586 i
5.41e-7 + 1.06e-6 i
Singularities of quadratic [9, 9, 8] approximant
2
-0.7579 - 0.6586 i
5.41e-7 - 1.06e-6 i
3
-0.7579 + 0.6586 i
1.06e-6 - 5.41e-7 i
4
-0.7579 - 0.6586 i
1.06e-6 + 5.41e-7 i
5
1.4076
0.00316
6
1.4067 + 0.2574 i
0.00813 - 0.00388 i
7
1.4067 - 0.2574 i
0.00813 + 0.00388 i
8
1.4326
0.00384 i
9
1.501 + 0.4338 i
0.013 - 0.013 i
10
1.501 - 0.4338 i
0.013 + 0.013 i
11
2.4356 + 0.9217 i
0.0822 + 0.0795 i
12
2.4356 - 0.9217 i
0.0822 - 0.0795 i
13
-3.3667
0.00513
14
3.8583
2.99
15
-4.2541
0.0174 i
16
-6.3191
0.0587
17
9.1342
1.32 i
18
-17.9853
29.7 i
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Table 20. Singularities with their weights for the quadratic approximant [9, 9, 9]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.7304 + 0.6768 i
7.13e-7 + 8.72e-7 i
Singularities of quadratic [9, 9, 9] approximant
2
-0.7304 - 0.6768 i
7.13e-7 - 8.72e-7 i
3
-0.7304 + 0.6768 i
8.72e-7 - 7.13e-7 i
4
-0.7304 - 0.6768 i
8.72e-7 + 7.13e-7 i
5
1.3947 + 0.0081 i
0.00165 - 0.00146 i
6
1.3947 - 0.0081 i
0.00165 + 0.00146 i
7
1.4072 + 0.2526 i
0.00731 - 0.00441 i
8
1.4072 - 0.2526 i
0.00731 + 0.00441 i
9
1.4969 + 0.4385 i
0.0121 - 0.0118 i
10
1.4969 - 0.4385 i
0.0121 + 0.0118 i
11
2.3694 + 0.9347 i
0.0775 + 0.0576 i
12
2.3694 - 0.9347 i
0.0775 - 0.0576 i
13
-3.4026
0.00631
14
3.4714
432.
15
-4.0517
0.0291 i
16
-5.6574
0.0266
17
10.6046
3.64 i
18
-13.8368
0.713 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

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