Singularities of Møller-Plesset series: example "a83"

Molecule HCl. 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
2.3493
0.688
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
2.906
1.07
Singularities of quadratic [1, 1, 0] approximant
2
230.9003
9.56 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
2.4929
0.614
Singularities of quadratic [1, 1, 1] approximant
2
16.2961
6.64 i
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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
1.1712
0.0799
Singularities of quadratic [2, 1, 1] approximant
2
1.3348
0.0874 i
3
4.7728
99.6
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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
-1.5267 + 0.1032 i
0.0244 + 0.0222 i
Singularities of quadratic [2, 2, 1] approximant
2
-1.5267 - 0.1032 i
0.0244 - 0.0222 i
3
1.9255
0.167
4
15.1405
27.8 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
-1.9157
0.0284
Singularities of quadratic [2, 2, 2] approximant
2
2.0405 + 1.1131 i
0.133 + 0.0199 i
3
2.0405 - 1.1131 i
0.133 - 0.0199 i
4
-3.3296
0.0378 i
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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
-2.1878
0.0484
Singularities of quadratic [3, 2, 2] approximant
2
2.7464 + 0.9806 i
0.423 + 0.19 i
3
2.7464 - 0.9806 i
0.423 - 0.19 i
4
-5.1815 + 1.9432 i
0.0276 - 0.0672 i
5
-5.1815 - 1.9432 i
0.0276 + 0.0672 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
0.7844
0.00101
Singularities of quadratic [3, 3, 2] approximant
2
0.7933
0.00102 i
3
-2.0367 + 0.7328 i
0.0154 + 0.00647 i
4
-2.0367 - 0.7328 i
0.0154 - 0.00647 i
5
2.103 + 1.8386 i
0.0362 - 0.0717 i
6
2.103 - 1.8386 i
0.0362 + 0.0717 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
-1.7104
0.00828
Singularities of quadratic [3, 3, 3] approximant
2
-1.2378 + 2.61 i
0.00702 + 0.013 i
3
-1.2378 - 2.61 i
0.00702 - 0.013 i
4
1.775 + 2.4598 i
0.00237 - 0.0412 i
5
1.775 - 2.4598 i
0.00237 + 0.0412 i
6
3.7629
0.299
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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
-1.8115 + 0.322 i
0.00614 + 0.00803 i
Singularities of quadratic [4, 3, 3] approximant
2
-1.8115 - 0.322 i
0.00614 - 0.00803 i
3
2.5281
0.873
4
-2.7489
0.0136
5
-0.9185 + 3.8385 i
0.098 - 0.021 i
6
-0.9185 - 3.8385 i
0.098 + 0.021 i
7
4.5356
1.44 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
1.1683 + 1.0809 i
0.0025 - 0.00101 i
Singularities of quadratic [4, 4, 3] approximant
2
1.1683 - 1.0809 i
0.0025 + 0.00101 i
3
1.2397 + 1.0335 i
0.0013 + 0.00256 i
4
1.2397 - 1.0335 i
0.0013 - 0.00256 i
5
-1.679
0.00619
6
-0.6793 + 2.912 i
0.0154 - 0.000603 i
7
-0.6793 - 2.912 i
0.0154 + 0.000603 i
8
-289.8786
2.41 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
-1.5887
0.00311
Singularities of quadratic [4, 4, 4] approximant
2
0.0913 + 1.669 i
0.000481 + 0.000603 i
3
0.0913 - 1.669 i
0.000481 - 0.000603 i
4
-0.0184 + 1.8631 i
0.000561 - 0.000587 i
5
-0.0184 - 1.8631 i
0.000561 + 0.000587 i
6
0.3441 + 1.9746 i
0.00136 + 0.0000108 i
7
0.3441 - 1.9746 i
0.00136 - 0.0000108 i
8
-3.224
0.0584 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
-1.5315
0.00209
Singularities of quadratic [5, 4, 4] approximant
2
0.0987 + 1.7337 i
0.000413 + 0.000744 i
3
0.0987 - 1.7337 i
0.000413 - 0.000744 i
4
-0.2402 + 1.8719 i
0.00101 - 0.000416 i
5
-0.2402 - 1.8719 i
0.00101 + 0.000416 i
6
0.4543 + 1.8409 i
0.00136 - 0.000716 i
7
0.4543 - 1.8409 i
0.00136 + 0.000716 i
8
-2.298
0.00838 i
9
-7.0949
0.0211
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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
-1.2074 + 0.0054 i
0.000249 + 0.000251 i
Singularities of quadratic [5, 5, 4] approximant
2
-1.2074 - 0.0054 i
0.000249 - 0.000251 i
3
-1.5984
0.00241
4
2.3929 + 0.571 i
0.0534 - 0.227 i
5
2.3929 - 0.571 i
0.0534 + 0.227 i
6
2.9661 + 1.5259 i
0.0201 - 0.184 i
7
2.9661 - 1.5259 i
0.0201 + 0.184 i
8
0.2311 + 3.5339 i
0.0288 - 0.0444 i
9
0.2311 - 3.5339 i
0.0288 + 0.0444 i
10
-11.6853
0.355 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
-1.0329 + 0.0013 i
0.0000954 + 0.0000955 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.0329 - 0.0013 i
0.0000954 - 0.0000955 i
3
-1.5887
0.00251
4
2.0245 + 1.1174 i
0.0287 + 0.00901 i
5
2.0245 - 1.1174 i
0.0287 - 0.00901 i
6
1.8545 + 1.3967 i
0.0106 - 0.0195 i
7
1.8545 - 1.3967 i
0.0106 + 0.0195 i
8
0.0901 + 3.0406 i
0.00895 - 0.0192 i
9
0.0901 - 3.0406 i
0.00895 + 0.0192 i
10
-5.314
8.76 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
-1.4764 + 0.0389 i
0.00079 + 0.00103 i
Singularities of quadratic [6, 5, 5] approximant
2
-1.4764 - 0.0389 i
0.00079 - 0.00103 i
3
-1.6257
0.00181
4
2.4762 + 0.5816 i
0.383 - 0.29 i
5
2.4762 - 0.5816 i
0.383 + 0.29 i
6
3.7381
1.05
7
-0.6092 + 4.3072 i
0.0846 + 0.048 i
8
-0.6092 - 4.3072 i
0.0846 - 0.048 i
9
2.7836 + 3.5692 i
0.313 + 0.513 i
10
2.7836 - 3.5692 i
0.313 - 0.513 i
11
-10.8656
0.103 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
-1.5355 + 0.0685 i
0.00111 + 0.00159 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.5355 - 0.0685 i
0.00111 - 0.00159 i
3
-1.6973
0.00232
4
2.4282 + 0.6585 i
0.216 - 0.159 i
5
2.4282 - 0.6585 i
0.216 + 0.159 i
6
-2.7463
0.145 i
7
-3.4008
0.0556
8
3.0953 + 1.6365 i
0.109 - 0.234 i
9
3.0953 - 1.6365 i
0.109 + 0.234 i
10
0.3563 + 3.8354 i
0.0524 - 0.0815 i
11
0.3563 - 3.8354 i
0.0524 + 0.0815 i
12
-10.5342
0.573 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
-1.5164 + 0.0475 i
0.00121 + 0.0017 i
Singularities of quadratic [6, 6, 6] approximant
2
-1.5164 - 0.0475 i
0.00121 - 0.0017 i
3
-1.674
0.00258
4
2.2363 + 0.7676 i
0.0965 - 0.0182 i
5
2.2363 - 0.7676 i
0.0965 + 0.0182 i
6
2.6172 + 1.0304 i
0.00726 - 0.134 i
7
2.6172 - 1.0304 i
0.00726 + 0.134 i
8
1.3993 + 3.5479 i
0.307 - 0.0666 i
9
1.3993 - 3.5479 i
0.307 + 0.0666 i
10
-1.4259 + 4.0714 i
0.0221 + 0.0514 i
11
-1.4259 - 4.0714 i
0.0221 - 0.0514 i
12
16.0583
1.93
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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
-1.5192 + 0.0463 i
0.00132 + 0.00186 i
Singularities of quadratic [7, 6, 6] approximant
2
-1.5192 - 0.0463 i
0.00132 - 0.00186 i
3
-1.6817
0.00281
4
2.2251 + 0.778 i
0.0869 - 0.00387 i
5
2.2251 - 0.778 i
0.0869 + 0.00387 i
6
2.6538 + 1.0272 i
0.0299 - 0.132 i
7
2.6538 - 1.0272 i
0.0299 + 0.132 i
8
1.4954 + 3.5304 i
0.278 - 0.249 i
9
1.4954 - 3.5304 i
0.278 + 0.249 i
10
-1.301 + 3.9891 i
0.0242 + 0.041 i
11
-1.301 - 3.9891 i
0.0242 - 0.041 i
12
3.1388 + 10.4519 i
0.209 + 0.0475 i
13
3.1388 - 10.4519 i
0.209 - 0.0475 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
-1.4918 + 0.054 i
0.000593 + 0.000883 i
Singularities of quadratic [7, 7, 6] approximant
2
-1.4918 - 0.054 i
0.000593 - 0.000883 i
3
-1.6066
0.00121
4
1.9657
0.013
5
2.1804 + 0.0947 i
0.00636 + 0.0175 i
6
2.1804 - 0.0947 i
0.00636 - 0.0175 i
7
2.4458 + 0.8194 i
0.0477 - 0.124 i
8
2.4458 - 0.8194 i
0.0477 + 0.124 i
9
1.065 + 3.7595 i
0.0959 + 0.0943 i
10
1.065 - 3.7595 i
0.0959 - 0.0943 i
11
-4.0566
0.0588 i
12
-5.3041 + 4.8052 i
0.335 + 0.0646 i
13
-5.3041 - 4.8052 i
0.335 - 0.0646 i
14
21.7421
1.52 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.