Singularities of Møller-Plesset series: example "a16"

Molecule BH. Basis cc-pVQZ. 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
1.2927
0.202
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
1.2872
0.2
Singularities of quadratic [1, 1, 0] approximant
2
287370.4764
94.7 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
1.5145
0.376
Singularities of quadratic [1, 1, 1] approximant
2
-8.5766
0.877
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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
1.4728
0.338
Singularities of quadratic [2, 1, 1] approximant
2
-4.1308
0.426
3
-6.9349
0.314 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
1.4469
0.294
Singularities of quadratic [2, 2, 1] approximant
2
-6.0145
1.91
3
-10.0817
0.705 i
4
168.9953
2.18 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
1.5145
0.502
Singularities of quadratic [2, 2, 2] approximant
2
3.1178
0.617 i
3
4.4759
11.
4
-5.4372
0.36
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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
1.3384 + 0.2217 i
0.00495 - 0.144 i
Singularities of quadratic [3, 2, 2] approximant
2
1.3384 - 0.2217 i
0.00495 + 0.144 i
3
1.3613
0.1
4
-4.9916
0.328
5
-20.3182
0.323 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
1.5711 + 0.3143 i
0.278 - 0.223 i
Singularities of quadratic [3, 3, 2] approximant
2
1.5711 - 0.3143 i
0.278 + 0.223 i
3
2.1419
0.501
4
-4.2286
0.157
5
-17.6661
0.22 i
6
45.1719
0.471 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
1.5988 + 0.3244 i
0.321 - 0.174 i
Singularities of quadratic [3, 3, 3] approximant
2
1.5988 - 0.3244 i
0.321 + 0.174 i
3
2.3332
0.828
4
-4.266
0.152
5
11.4301
0.49 i
6
-224.8279
0.111 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
1.6049 + 0.3342 i
0.319 - 0.128 i
Singularities of quadratic [4, 3, 3] approximant
2
1.6049 - 0.3342 i
0.319 + 0.128 i
3
2.402
1.11
4
-4.159
0.127
5
13.8183
0.448 i
6
-0.5204 + 17.4617 i
0.0733 + 0.274 i
7
-0.5204 - 17.4617 i
0.0733 - 0.274 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
1.6043 + 0.3399 i
0.314 - 0.0993 i
Singularities of quadratic [4, 4, 3] approximant
2
1.6043 - 0.3399 i
0.314 + 0.0993 i
3
2.3637
1.07
4
-3.1665
0.03
5
-3.544
0.0319 i
6
-6.5087 + 2.4242 i
0.062 - 0.198 i
7
-6.5087 - 2.4242 i
0.062 + 0.198 i
8
63.3988
0.602 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
1.6 + 0.3118 i
0.315 - 0.249 i
Singularities of quadratic [4, 4, 4] approximant
2
1.6 - 0.3118 i
0.315 + 0.249 i
3
-1.9444 + 0.0014 i
0.00402 + 0.00403 i
4
-1.9444 - 0.0014 i
0.00402 - 0.00403 i
5
2.3916
0.817
6
-4.0796
0.0955
7
6.0118
0.794 i
8
14.2009
7.06
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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
-0.2487 + 0.e-5 i
2.81e-6 + 2.81e-6 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.2487 - 0.e-5 i
2.81e-6 - 2.81e-6 i
3
1.5974 + 0.3079 i
0.287 - 0.272 i
4
1.5974 - 0.3079 i
0.287 + 0.272 i
5
2.4255
0.888
6
-4.0841
0.105
7
6.6919
0.556 i
8
10.9588 + 16.2174 i
0.154 + 0.424 i
9
10.9588 - 16.2174 i
0.154 - 0.424 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
1.2265 + 0.0048 i
0.0213 - 0.0215 i
Singularities of quadratic [5, 5, 4] approximant
2
1.2265 - 0.0048 i
0.0213 + 0.0215 i
3
1.6418 + 0.3752 i
0.32 + 0.12 i
4
1.6418 - 0.3752 i
0.32 - 0.12 i
5
2.3689
1.49
6
-3.1548
0.043
7
-3.3424
0.0446 i
8
-5.5501
123.
9
-9.8609
0.22 i
10
82.8276
0.694 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
1.2934 + 0.0093 i
0.03 - 0.0305 i
Singularities of quadratic [5, 5, 5] approximant
2
1.2934 - 0.0093 i
0.03 + 0.0305 i
3
1.6559 + 0.3853 i
0.293 + 0.18 i
4
1.6559 - 0.3853 i
0.293 - 0.18 i
5
2.3776
1.81
6
-3.5435
0.0491
7
-4.7537
0.0539 i
8
-4.982 + 3.2699 i
0.0712 - 0.112 i
9
-4.982 - 3.2699 i
0.0712 + 0.112 i
10
-22.3706
4.41
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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
1.3937 + 0.0365 i
0.0338 - 0.0298 i
Singularities of quadratic [6, 5, 5] approximant
2
1.3937 - 0.0365 i
0.0338 + 0.0298 i
3
1.7322 + 0.4588 i
0.0791 - 0.279 i
4
1.7322 - 0.4588 i
0.0791 + 0.279 i
5
2.079
1.29
6
-3.0593 + 0.0721 i
0.017 + 0.0172 i
7
-3.0593 - 0.0721 i
0.017 - 0.0172 i
8
-4.3937
0.0964
9
4.814
36.5 i
10
7.3289
0.63
11
-11.1419
0.48 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
1.4408 + 0.065 i
0.0492 - 0.0375 i
Singularities of quadratic [6, 6, 5] approximant
2
1.4408 - 0.065 i
0.0492 + 0.0375 i
3
1.6892 + 0.5332 i
0.0325 - 0.148 i
4
1.6892 - 0.5332 i
0.0325 + 0.148 i
5
2.0631 + 0.2204 i
0.735 + 0.0419 i
6
2.0631 - 0.2204 i
0.735 - 0.0419 i
7
3.0388
134.
8
-3.2432
0.208
9
-3.3141
0.205 i
10
-5.164
0.922
11
-9.9841
0.29 i
12
156.5158
1.06 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
1.5753 + 0.1598 i
0.923 - 3.46 i
Singularities of quadratic [6, 6, 6] approximant
2
1.5753 - 0.1598 i
0.923 + 3.46 i
3
1.5774 + 0.4361 i
0.159 - 0.0536 i
4
1.5774 - 0.4361 i
0.159 + 0.0536 i
5
1.6759 + 0.172 i
0.0187 - 0.195 i
6
1.6759 - 0.172 i
0.0187 + 0.195 i
7
1.9546
0.172
8
-3.1416
0.0334
9
-3.3477
0.0359 i
10
-5.219
5.52
11
17.7145
0.469 i
12
-17.843
0.176 i
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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
1.4616
0.0494
Singularities of quadratic [7, 6, 6] approximant
2
1.4751 + 0.3218 i
0.0262 + 0.00637 i
3
1.4751 - 0.3218 i
0.0262 - 0.00637 i
4
1.4641 + 0.4295 i
0.0134 - 0.0205 i
5
1.4641 - 0.4295 i
0.0134 + 0.0205 i
6
1.9438 + 0.5333 i
0.0706 + 0.118 i
7
1.9438 - 0.5333 i
0.0706 - 0.118 i
8
-3.1049
0.0132
9
-3.5762
0.016 i
10
-4.1153 + 7.9739 i
0.000498 - 0.0628 i
11
-4.1153 - 7.9739 i
0.000498 + 0.0628 i
12
-10.8104 + 2.4534 i
0.0385 + 0.0328 i
13
-10.8104 - 2.4534 i
0.0385 - 0.0328 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
1.4253 + 0.2636 i
0.0302 - 0.0184 i
Singularities of quadratic [7, 7, 6] approximant
2
1.4253 - 0.2636 i
0.0302 + 0.0184 i
3
1.485 + 0.3406 i
0.0159 + 0.0437 i
4
1.485 - 0.3406 i
0.0159 - 0.0437 i
5
1.5298
0.0716
6
1.8707 + 0.4191 i
0.214 + 0.147 i
7
1.8707 - 0.4191 i
0.214 - 0.147 i
8
-3.1509
0.0308
9
-3.3832
0.0335 i
10
-5.3578
28.1
11
6.8793
1.09 i
12
9.9577
6.69
13
-13.295
0.199 i
14
153.6699
0.97 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

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