Singularities of Møller-Plesset series: example "a84"

Molecule HCl. Basis aug-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
2.4492
0.78
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
2.1545
0.607
Singularities of quadratic [1, 1, 0] approximant
2
558.8694
9.78 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
3.8749
16.8
Singularities of quadratic [1, 1, 1] approximant
2
-4.8952
0.672
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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
-1.7267
0.473
Singularities of quadratic [2, 1, 1] approximant
2
-1.933
0.436 i
3
2.5662
1.03
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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
-2.051 + 0.1945 i
0.273 + 0.217 i
Singularities of quadratic [2, 2, 1] approximant
2
-2.051 - 0.1945 i
0.273 - 0.217 i
3
2.3644
0.616
4
258.5629
12.5 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
-1.869
0.182
Singularities of quadratic [2, 2, 2] approximant
2
2.5719
1.43
3
-2.637
0.188 i
4
11.917
2.1 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
-1.7196
0.0782
Singularities of quadratic [3, 2, 2] approximant
2
1.8538 + 0.6328 i
0.0644 - 0.0892 i
3
1.8538 - 0.6328 i
0.0644 + 0.0892 i
4
2.0596
0.0877
5
-2.8881
0.115 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
-1.4338
0.0197
Singularities of quadratic [3, 3, 2] approximant
2
-1.7521 + 1.9354 i
0.00943 - 0.0365 i
3
-1.7521 - 1.9354 i
0.00943 + 0.0365 i
4
3.0553
81.5
5
-3.8327
0.0367 i
6
13.5609
2.53 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
-1.4553
0.0202
Singularities of quadratic [3, 3, 3] approximant
2
-1.2877 + 2.3824 i
0.00624 + 0.0328 i
3
-1.2877 - 2.3824 i
0.00624 - 0.0328 i
4
3.0218
38.
5
-0.3547 + 5.1451 i
0.0947 - 0.00639 i
6
-0.3547 - 5.1451 i
0.0947 + 0.00639 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
-0.2243 + 0.e-5 i
5.28e-6 + 5.28e-6 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.2243 - 0.e-5 i
5.28e-6 - 5.28e-6 i
3
-1.4127
0.0129
4
2.514
0.538
5
-1.4105 + 3.2663 i
0.0618 + 0.0338 i
6
-1.4105 - 3.2663 i
0.0618 - 0.0338 i
7
11.463
0.955 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
-0.9227 + 0.0034 i
0.00048 + 0.000482 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.9227 - 0.0034 i
0.00048 - 0.000482 i
3
-1.3614
0.00655
4
2.292
0.197
5
4.3415
539. i
6
-0.0877 + 5.4501 i
0.194 - 0.13 i
7
-0.0877 - 5.4501 i
0.194 + 0.13 i
8
-9.428
1.07 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
-1.0709 + 0.0103 i
0.000922 + 0.000948 i
Singularities of quadratic [4, 4, 4] approximant
2
-1.0709 - 0.0103 i
0.000922 - 0.000948 i
3
-1.358
0.0054
4
2.3663
0.304
5
0.5267 + 6.0192 i
0.471 - 0.188 i
6
0.5267 - 6.0192 i
0.471 + 0.188 i
7
6.2299
4.01 i
8
-25.6163
0.379 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
0.3478
2.35e-6
Singularities of quadratic [5, 4, 4] approximant
2
0.3478
2.35e-6 i
3
-1.2952
0.00658
4
-1.5354
0.0106 i
5
-1.8536
2.
6
2.2025
0.0936
7
-1.0721 + 3.4952 i
0.0489 + 0.016 i
8
-1.0721 - 3.4952 i
0.0489 - 0.016 i
9
4.3498
18.5 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
0.1639 + 0.e-4 i
7.66e-8 - 7.66e-8 i
Singularities of quadratic [5, 5, 4] approximant
2
0.1639 - 0.e-4 i
7.66e-8 + 7.66e-8 i
3
-1.2892
0.00658
4
-1.4877
0.0103 i
5
-1.7626
4.09
6
2.2265
0.112
7
-1.0724 + 3.7031 i
0.0609 + 0.0147 i
8
-1.0724 - 3.7031 i
0.0609 - 0.0147 i
9
4.427
18.5 i
10
-484.7609
12.1 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
-1.2161 + 0.0354 i
0.00139 + 0.00173 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.2161 - 0.0354 i
0.00139 - 0.00173 i
3
-1.3552
0.00335
4
2.1273
0.0892
5
2.7364
0.241 i
6
0.805 + 3.5147 i
0.102 + 0.042 i
7
0.805 - 3.5147 i
0.102 - 0.042 i
8
-1.2932 + 5.6866 i
0.0508 - 0.105 i
9
-1.2932 - 5.6866 i
0.0508 + 0.105 i
10
18.6399
1.57
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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
-1.2057 + 0.0323 i
0.00127 + 0.00155 i
Singularities of quadratic [6, 5, 5] approximant
2
-1.2057 - 0.0323 i
0.00127 - 0.00155 i
3
-1.3456
0.00316
4
2.1045
0.0914
5
2.5036
0.163 i
6
4.0711
1.05
7
0.536 + 4.2198 i
0.196 + 0.276 i
8
0.536 - 4.2198 i
0.196 - 0.276 i
9
-0.6956 + 5.4464 i
0.423 - 0.332 i
10
-0.6956 - 5.4464 i
0.423 + 0.332 i
11
-6.0175
0.18 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
-1.1131 + 0.0151 i
0.000365 + 0.000393 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.1131 - 0.0151 i
0.000365 - 0.000393 i
3
-1.2589
0.00136
4
2.0863
0.0537
5
-2.3962
7.75 i
6
2.9265
0.339 i
7
-3.0142 + 1.0608 i
0.0292 - 0.0615 i
8
-3.0142 - 1.0608 i
0.0292 + 0.0615 i
9
0.4074 + 3.2627 i
0.00534 + 0.0369 i
10
0.4074 - 3.2627 i
0.00534 - 0.0369 i
11
3.5692 + 4.1724 i
0.156 - 0.1 i
12
3.5692 - 4.1724 i
0.156 + 0.1 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
1.0925
0.0000714
Singularities of quadratic [6, 6, 6] approximant
2
1.0928
0.0000714 i
3
-1.1616 + 0.0283 i
0.000509 + 0.000585 i
4
-1.1616 - 0.0283 i
0.000509 - 0.000585 i
5
-1.2814
0.00138
6
2.1028 + 0.3597 i
0.0178 + 0.0065 i
7
2.1028 - 0.3597 i
0.0178 - 0.0065 i
8
1.1277 + 2.9035 i
0.0179 - 0.0135 i
9
1.1277 - 2.9035 i
0.0179 + 0.0135 i
10
-1.8079 + 3.2767 i
0.0188 - 0.0277 i
11
-1.8079 - 3.2767 i
0.0188 + 0.0277 i
12
3.9399
0.595
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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
0.1363
4.31e-10 - 4.31e-10 i
Singularities of quadratic [7, 6, 6] approximant
2
0.1363
4.31e-10 + 4.31e-10 i
3
-1.1565 + 0.0287 i
0.000435 + 0.000495 i
4
-1.1565 - 0.0287 i
0.000435 - 0.000495 i
5
-1.2724
0.00118
6
2.2738 + 0.1979 i
0.115 + 0.0128 i
7
2.2738 - 0.1979 i
0.115 - 0.0128 i
8
1.1475 + 2.88 i
0.018 - 0.0183 i
9
1.1475 - 2.88 i
0.018 + 0.0183 i
10
-1.6856 + 3.3603 i
0.0101 - 0.0309 i
11
-1.6856 - 3.3603 i
0.0101 + 0.0309 i
12
5.8148 + 4.3608 i
0.135 + 0.249 i
13
5.8148 - 4.3608 i
0.135 - 0.249 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
-1.1061
0.000177
Singularities of quadratic [7, 7, 6] approximant
2
-1.1479 + 0.0737 i
0.000111 - 0.000128 i
3
-1.1479 - 0.0737 i
0.000111 + 0.000128 i
4
-1.1829 + 0.0806 i
0.000107 + 0.000187 i
5
-1.1829 - 0.0806 i
0.000107 - 0.000187 i
6
2.2009 + 0.2876 i
0.0527 - 0.00426 i
7
2.2009 - 0.2876 i
0.0527 + 0.00426 i
8
1.857 + 2.6393 i
0.0295 + 0.0229 i
9
1.857 - 2.6393 i
0.0295 - 0.0229 i
10
3.1891 + 1.5261 i
0.0336 + 0.114 i
11
3.1891 - 1.5261 i
0.0336 - 0.114 i
12
-0.6052 + 3.7825 i
0.0399 + 0.0191 i
13
-0.6052 - 3.7825 i
0.0399 - 0.0191 i
14
-16.5383
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

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.