Singularities of Møller-Plesset series: example "a44"

Molecule BO+. 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, 1, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.3684
0.0571
Singularities of quadratic [1, 1, 0] approximant
2
-0.8976
0.0892 i
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Table 2. 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
-0.6746
0.212
Singularities of quadratic [1, 1, 1] approximant
2
8.4475
1.57
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Table 3. 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
-0.4915
0.0542
Singularities of quadratic [2, 1, 1] approximant
2
0.7452 + 1.0275 i
0.103 + 0.00756 i
3
0.7452 - 1.0275 i
0.103 - 0.00756 i
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Table 4. 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
-0.5089
0.0646
Singularities of quadratic [2, 2, 1] approximant
2
0.9609 + 1.0451 i
0.135 + 0.0076 i
3
0.9609 - 1.0451 i
0.135 - 0.0076 i
4
-64.8418
3.84 i
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Table 5. 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
0.0719
0.0000567
Singularities of quadratic [2, 2, 2] approximant
2
0.0719
0.0000567 i
3
-0.4707
0.0329
4
1.2306
0.121
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Table 6. 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
0.2387 + 0.0002 i
0.000526 - 0.000525 i
Singularities of quadratic [3, 2, 2] approximant
2
0.2387 - 0.0002 i
0.000526 + 0.000525 i
3
-0.4833
0.0419
4
1.5107
0.236
5
4.3914
4.36 i
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Table 7. 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
-0.4859
0.0448
Singularities of quadratic [3, 3, 2] approximant
2
0.5252 + 0.0064 i
0.00309 - 0.00305 i
3
0.5252 - 0.0064 i
0.00309 + 0.00305 i
4
1.3395
0.0918
5
3.9881
90.3 i
6
-140.185
13.2 i
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Table 8. 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
-0.4867
0.0479
Singularities of quadratic [3, 3, 3] approximant
2
-0.1053 + 0.6566 i
0.00849 - 0.0148 i
3
-0.1053 - 0.6566 i
0.00849 + 0.0148 i
4
-0.128 + 0.6598 i
0.0144 + 0.00884 i
5
-0.128 - 0.6598 i
0.0144 - 0.00884 i
6
1.388
0.273
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Table 9. 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
-0.3605 + 0.0006 i
0.00188 + 0.00189 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.3605 - 0.0006 i
0.00188 - 0.00189 i
3
-0.4733
0.0252
4
1.3289
0.286
5
0.0714 + 2.1233 i
0.0207 + 0.27 i
6
0.0714 - 2.1233 i
0.0207 - 0.27 i
7
-14.3235
1.2 i
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Table 10. 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
-0.0406
2.41e-7
Singularities of quadratic [4, 4, 3] approximant
2
-0.0406
2.41e-7 i
3
-0.4755
0.0317
4
1.2025
0.0986
5
-1.5307
0.507 i
6
-1.9408 + 1.7912 i
0.409 + 0.863 i
7
-1.9408 - 1.7912 i
0.409 - 0.863 i
8
11.0478
499. i
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Table 11. 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
-0.502
0.131
Singularities of quadratic [4, 4, 4] approximant
2
-0.5608
0.331 i
3
-0.6432
0.263
4
-1.1901
0.264 i
5
1.2658 + 0.1781 i
0.109 - 0.102 i
6
1.2658 - 0.1781 i
0.109 + 0.102 i
7
1.7864
0.285
8
-2.2802
832.
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Table 12. 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
-0.5093
0.219
Singularities of quadratic [5, 4, 4] approximant
2
-0.5399
1.58 i
3
-0.6224
0.171
4
-1.2326
0.261 i
5
1.2629 + 0.1703 i
0.108 - 0.106 i
6
1.2629 - 0.1703 i
0.108 + 0.106 i
7
1.752
0.27
8
-2.488
97.7
9
-34.9609
0.883 i
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Table 13. 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
-0.5158 + 0.0348 i
0.089 + 0.0382 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.5158 - 0.0348 i
0.089 - 0.0382 i
3
-0.6168
0.238
4
-0.4255 + 0.6014 i
0.00521 - 0.00544 i
5
-0.4255 - 0.6014 i
0.00521 + 0.00544 i
6
-0.4137 + 0.614 i
0.00554 + 0.00507 i
7
-0.4137 - 0.614 i
0.00554 - 0.00507 i
8
1.1637
0.0781
9
4.9348
7.03 i
10
-5.2832
9.18 i
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Table 14. 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
0.1005
3.23e-7
Singularities of quadratic [5, 5, 5] approximant
2
0.1005
3.23e-7 i
3
-0.5172
1.08
4
0.5312
0.00045
5
0.5327
0.000451 i
6
-0.5384
0.643 i
7
-0.6385
0.336
8
1.0829
0.0279
9
-1.5288
0.196 i
10
-4.6261
0.898
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Table 15. 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
-0.5163 + 0.0213 i
0.0427 + 0.212 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.5163 - 0.0213 i
0.0427 - 0.212 i
3
-0.5987
0.114
4
0.9051 + 0.7127 i
0.0136 - 0.0259 i
5
0.9051 - 0.7127 i
0.0136 + 0.0259 i
6
1.1611
0.0685
7
0.8977 + 0.7623 i
0.0273 + 0.0151 i
8
0.8977 - 0.7623 i
0.0273 - 0.0151 i
9
-1.4093
0.223 i
10
-4.2368 + 2.6667 i
0.288 + 0.732 i
11
-4.2368 - 2.6667 i
0.288 - 0.732 i
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Table 16. 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
0.4349 + 0.e-4 i
0.000137 - 0.000137 i
Singularities of quadratic [6, 6, 5] approximant
2
0.4349 - 0.e-4 i
0.000137 + 0.000137 i
3
-0.5206 + 0.0187 i
0.165 + 0.25 i
4
-0.5206 - 0.0187 i
0.165 - 0.25 i
5
-0.6158
0.176
6
1.4456 + 0.1362 i
0.161 + 0.238 i
7
1.4456 - 0.1362 i
0.161 - 0.238 i
8
-1.8967
0.196 i
9
-0.3142 + 2.2012 i
0.0564 + 0.0849 i
10
-0.3142 - 2.2012 i
0.0564 - 0.0849 i
11
1.1813 + 2.0801 i
0.0122 - 0.113 i
12
1.1813 - 2.0801 i
0.0122 + 0.113 i
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Table 17. 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
0.1106
1.07e-7
Singularities of quadratic [6, 6, 6] approximant
2
0.1106
1.07e-7 i
3
-0.5234 + 0.0173 i
0.293 + 0.195 i
4
-0.5234 - 0.0173 i
0.293 - 0.195 i
5
-0.6275
0.266
6
1.1932 + 0.2526 i
0.0546 - 0.023 i
7
1.1932 - 0.2526 i
0.0546 + 0.023 i
8
-1.3844 + 1.09 i
0.0607 + 0.105 i
9
-1.3844 - 1.09 i
0.0607 - 0.105 i
10
-0.9386 + 1.5147 i
0.0386 - 0.108 i
11
-0.9386 - 1.5147 i
0.0386 + 0.108 i
12
1.7889
0.22
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Table 18. 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
-0.523 + 0.0132 i
0.192 + 0.471 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.523 - 0.0132 i
0.192 - 0.471 i
3
-0.6467
0.3
4
-0.7566
1.72 i
5
-0.8581
0.49
6
1.2494 + 0.2224 i
0.0866 - 0.089 i
7
1.2494 - 0.2224 i
0.0866 + 0.089 i
8
-1.2847
0.368 i
9
1.9754 + 0.5013 i
0.299 + 0.148 i
10
1.9754 - 0.5013 i
0.299 - 0.148 i
11
-1.8867 + 1.5953 i
0.881 + 0.0895 i
12
-1.8867 - 1.5953 i
0.881 - 0.0895 i
13
6.9332
0.957
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Table 19. 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
-0.5227 + 0.0131 i
0.142 + 0.486 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.5227 - 0.0131 i
0.142 - 0.486 i
3
-0.6463
0.274
4
-0.7442
4.4 i
5
-0.8385
0.373
6
1.2482 + 0.2268 i
0.087 - 0.0774 i
7
1.2482 - 0.2268 i
0.087 + 0.0774 i
8
-1.301
0.381 i
9
2.0444 + 0.554 i
0.303 + 0.254 i
10
2.0444 - 0.554 i
0.303 - 0.254 i
11
-1.9223 + 1.4827 i
1.02 + 0.481 i
12
-1.9223 - 1.4827 i
1.02 - 0.481 i
13
11.4624
2.14
14
86.4314
73.7 i
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Table 20. 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
-0.2259
2.74e-6
Singularities of quadratic [7, 7, 7] approximant
2
-0.2259
2.74e-6 i
3
-0.5295 + 0.0161 i
0.238 - 0.14 i
4
-0.5295 - 0.0161 i
0.238 + 0.14 i
5
-0.6551
1.62
6
-1.0576
0.216 i
7
-1.1923 + 0.3366 i
0.0232 + 0.142 i
8
-1.1923 - 0.3366 i
0.0232 - 0.142 i
9
1.2431 + 0.2563 i
0.0781 - 0.0214 i
10
1.2431 - 0.2563 i
0.0781 + 0.0214 i
11
2.9548 + 0.9199 i
0.715 + 0.514 i
12
2.9548 - 0.9199 i
0.715 - 0.514 i
13
-1.1628 + 2.9366 i
0.137 + 0.174 i
14
-1.1628 - 2.9366 i
0.137 - 0.174 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.